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D.V. Ionescu, Institutul de Calcul
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D.V. Ionescu, Integrarea unei ecuații diferențiale (Romanian) Acad. R. P. Romîne. Fil. Cluj. Stud. Cerc. Mat. 8 1957 275–289.
[Integration of a differential equation]
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Studii si Cercetari Matematice
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Academy of the Republic of S.R.
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1220-269X
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INTEGRATION OF A DIFFERENTIAL EQUATION
In this paper we will deal with the integration of the differential equation
{:(1)Delta_(n)[y]=|[y,y^('),dots,y^((n))],[y^('),y^(''),dots,y^((n+1))],[*,*,*,*],[y^((n)),y^((n+1)),dots,y^((2n))]|=0:}\Delta_{n}[y]=\left|\begin{array}{cccc} y & y^{\prime} & \ldots & y^{(n)} \tag{1}\\ y^{\prime} & y^{\prime \prime} & \ldots & y^{(n+1)} \\ \cdot & \cdot & \cdot & \cdot \\ y^{(n)} & \ldots & y^{(n+1)} \end{array}\right|=0
This problem was posed by Prof. T. Popoviciu during a paper given at the Institute of Computing in Cluj regarding a paper by H. Lö w ne r on monotone matrix functions [1].
We will also integrate the differential equation
{:(2)Delta_(n)[y]=Ae^(alpha x):}\begin{equation*} \Delta_{n}[y]=A e^{\alpha x} \tag{2} \end{equation*}
whereAAandalpha\alphaare constants.
We recall that
We recall that
{:(3)Delta_(1)[y]=1:}\begin{equation*} \Delta_{1}[y]=1 \tag{3} \end{equation*}
is the equation of the chain and that the equation
{:(4)Delta_(2)[y]=1:}\begin{equation*} \Delta_{2}[y]=1 \tag{4} \end{equation*}
was studied by G. Darboux [2]. Indications for the integration of the differential equation (2) were given by G. Darboux [3]. We will not follow the method given by G. Darboux, but will give a direct method for the integration of equations (1) and (2), which we will then apply to the particular cases (3) and (4.)
- Whetheryyan integral of the differential equationDelta_(n)(y)\Delta_{n}(y)continuous and with successive derivatives continuous up to the order2n2 nincluding within an interval (alpha,beta\alpha, \beta), which can be(-oo,+oo)(-\infty,+\infty)It can identically cancel the coefficient ofy^((2n))y^{(2 n)}from the equationDelta_(n)[y]=0\Delta_{n}[y]=0, that is, onDelta_(n-1)[y]\Delta_{n-1}[y], or not. We will deal further with the first case and now consider the case whenDelta_(n-1)[y]\Delta_{n-1}[y]is not identically null in the interval(alpha,beta)(\alpha, \beta). Then for a pointx_(0)x_{0}from this interval, we haveDelta_(n-1)[y(x_(0))]!=0\Delta_{n-1}\left[y\left(x_{0}\right)\right] \neq 0andDelta_(n-1)[y]\Delta_{n-1}[y]being a continuous function ofxxin the interval(alpha,beta)(\alpha, \beta)
, an interval can be determined.(a,b)(a, b)included in the range(alpha,beta)(\alpha, \beta)and which contains the pointx_(0)x_{0}, for whichDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0We will therefore place ourselves in the interval (a,ba, b) and we will determine the integralsyyof the differential equationDelta_(n)[y]=0\Delta_{n}[y]=0, which are such thatDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0Once these integrals are determined, we will show that the intervals(alpha,beta)(\alpha, \beta)and(a,b)(a, b)can be extended to the range(-oo,+oo)(-\infty,+\infty).
Let us determine for such an integralyy, functionslambda_(0)(x),lambda_(1)(x)dots,lambda_(n-1)(x)\lambda_{0}(x), \lambda_{1}(x) \ldots, \lambda_{n-1}(x)through linear equations
{:(5){:[lambda_(0)(x)y+lambda_(1)(x)y^(')+dots+lambda_(n-1)(x)y^((n-1))+y^((n))=0],[lambda_(0)(x)y^(')+lambda_(1)(x)y^('')+dots+ lambda_(n-1)(x)y^((n))+y^((n+1))=0],[lambda_(0)(x)y^((n-1))+lambda_(1)(x)y^((n))+dots+lambda_(n-1)(x)y^((2n-2))+y^((2n-1))=0]:}:}\begin{array}{lll} \lambda_{0}(x) y+\lambda_{1}(x) y^{\prime}+\ldots+\lambda_{n-1}(x) y^{(n-1)}+y^{(n)}=0 \\ \lambda_{0}(x) y^{\prime}+\lambda_{1}(x) y^{\prime}+\lambda_{n-1}(x) y^{(n-1)}+y^{(n)}=0 y^{(n)}+y^{(n+1)}=0 \tag{5}\\ \lambda_{0}(x) y^{(n-1)}+\lambda_{1}(x) y^{(n)}+\ldots+\lambda_{n-1}(x) y^{(2 n-2)}+y^{(2 n-1)}=0 \end{array}
This is possible because the determinant of the system isDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0Functionslambda_(0)(x),lambda_(1)(x),dots,lambda_(n-1)(x)\lambda_{0}(x), \lambda_{1}(x), \ldots, \lambda_{n-1}(x)are also differentiable. Because of the differential equation (1), we can add to the previous equations the equation
{:('")"lambda_(0)(x)y^((n))+lambda_(1)(x)y^((n+1))+dots+lambda_(n-1)(x)y^((2n-1))+y^((2n))=0:}\begin{equation*} \lambda_{0}(x) y^{(n)}+\lambda_{1}(x) y^{(n+1)}+\ldots+\lambda_{n-1}(x) y^{(2 n-1)}+y^{(2 n)}=0 \tag{$\prime$} \end{equation*}
Deriving each equation (5) in turn and taking into account the next equation, the last one being (5^(')5^{\prime}), the equations are derived
{:[lambda_(0)^(')(x)y+lambda_(1)^(')(x)y^(')+dots+lambda_(n-1)^(')(x)y^((n-1))=0],[lambda_(0)^(')(x)y^(')+lambda_(1)^(')(x)y^('' )+dots+lambda_(n-1)^(')(x)y^((n))=0],[lambda_(0)^(')(x)y^((n-1))+lambda_(1)^(')(x)y^((n))+dots+lambda_(n-1)^(')(x)y^((2n-2))=0]:}\begin{aligned} & \lambda_{0}^{\prime}(x) y+\lambda_{1}^{\prime}(x) y^{\prime}+\ldots+\lambda_{n-1}^{\prime}(x) y^{(n-1)}=0 \\ & \lambda_{0}^{\prime}(x) y^{\prime}(x) \prime}+\ldots+\lambda_{n-1}^{\prime}(x) y^{(n)}=0 \\ & \lambda_{0}^{\prime}(x) y^{(n-1)}+\lambda_{1}^{\prime}(x) y^{(n)}+\ldots+\lambda_{n-1}^{\prime}(x) y^{(2 n-2)}=0 \end{aligned}
linear and homogeneous inlambda_(0)^(')(x),lambda_(1)^(')(x),dots,lambda_(n-1)^(')(x)\lambda_{0}^{\prime}(x), \lambda_{1}^{\prime}(x), \ldots, \lambda_{n-1}^{\prime}(x)with the determinantDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0It follows that
lambda_(0)^(')(x)=0,quadlambda_(1)^(')(x)=0,dots,quadlambda_(n-1)^(')(x)=0\lambda_{0}^{\prime}(x)=0, \quad \lambda_{1}^{\prime}(x)=0, \ldots, \quad \lambda_{n-1}^{\prime}(x)=0
MEAN
lambda_(0)(x)=A_(n),quadlambda_(1)(x)=A_(n-1),dots,quadlambda_(n-1)(x)=A_(1)\lambda_{0}(x)=A_{n}, \quad \lambda_{1}(x)=A_{n-1}, \ldots, \quad \lambda_{n-1}(x)=A_{1}
A_(1),A_(2),dots,A_(n)A_{1}, A_{2}, \ldots, A_{n}being constants attached to the functiony(x)y(x); doing in the system of equations (5)x=x_(0)x=x_{0}, we observe thatA_(1),A_(2),dots,A_(n)A_{1}, A_{2}, \ldots, A_{n}, are given by the equations
{:[A_(n)y_(0)+A_(n-1)y_(0)^(')+dots+A_(1)y_(0)^((n-1))+y_(0)^((n))=0],[A_(n)y_(0)^(')+A_(n-1)y_(0)^('')+dots+A_(1)y_(0)^((n))+y_(0)^((n+1))=0],[(6)A_(n)y_(0)^((n-1))+A_(n-1)y_(0)^((n))+dots+A_(1)y_(0)^((2n-2))+y_(0)^((2n-1))=0]:}\begin{align*}
& A_{n} y_{0}+A_{n-1} y_{0}^{\prime}+\ldots+A_{1} y_{0}^{(n-1)}+y_{0}^{(n)}=0 \\
& A_{n} y_{0}^{\prime}+A_{n-1} y_{0}^{\prime \prime}+\ldots+A_{1} y_{0}^{(n)}+y_{0}^{(n+1)}=0 \\
& A_{n} y_{0}^{(n-1)}+A_{n-1} y_{0}^{(n)}+\ldots+A_{1} y_{0}^{(2 n-2)}+y_{0}^{(2 n-1)}=0 \tag{6}
\end{align*}
where
{:(7)y^((k))(x_(0))=y_(0)^((k))quad(k=0","1","2","dots","2n-1):}\begin{equation*}
y^{(k)}\left(x_{0}\right)=y_{0}^{(k)} \quad(k=0,1,2, \ldots, 2 n-1) \tag{7}
\end{equation*}
are Cauchy's conditions.
It follows that any integral of the differential equation (1) for whichDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0, is the integral of the differential equation
It follows that any integral of the differential equation (1) for whichDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0, is the integral of the differential equation
{:(8)y^((n))+A_(1)y^((n-1))+dots+A_(n)y=0:}\begin{equation*}
y^{(n)}+A_{1} y^{(n-1)}+\ldots+A_{n} y=0 \tag{8}
\end{equation*}
with constant coefficientsA_(1),A_(2),dots,A_(n)A_{1}, A_{2}, \ldots, A_{n}determined by the system of linear equations (6) by Cauchy's conditions.
Conversely, any integral of the equation with constant coefficients (8), for whichDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0, is the integral of the differential equation (1), which is immediately proven.
2. The characteristic equation of the differential equation (8) is obtained by eliminatingA_(1),A_(2),dots,A_(n)A_{1}, A_{2}, \ldots, A_{n}, between equations (6) and equation
2. The characteristic equation of the differential equation (8) is obtained by eliminatingA_(1),A_(2),dots,A_(n)A_{1}, A_{2}, \ldots, A_{n}, between equations (6) and equation
{:(9)r^(n)+A_(1)r^(n-1)+dots+A_(n)=0:}\begin{equation*}
r^{n}+A_{1} r^{n-1}+\ldots+A_{n}=0 \tag{9}
\end{equation*}
This is possible because it was assumedDelta_(n-1)[y(x_(0))]!=0\Delta_{n-1}\left[y\left(x_{0}\right)\right] \neq 0The characteristic equation is therefore
{:('")"varphi(r)=|[1,r,r^(2),dots,r^(n)],[y_(0),y_(0)^('),y_(0)^(''),dots,y_(0)^((n))],[y_(0)^('),y_(0)^(''),y_(0)^('''),dots,y_(0)^((n+1))],[*,*,*,*,*],[*,*,*],[y_(0)^((n-1)),y_(0)^((n)),y_(0)^((n+1)),dots,y_(0)^((2n-1))]|=0:}\varphi(r)=\left|\begin{array}{ccccc}
1 & r & r^{2} & \ldots & r^{n} \tag{$\prime$}\\
y_{0} & y_{0}^{\prime} & y_{0}^{\prime \prime} & \ldots & y_{0}^{(n)} \\
y_{0}^{\prime} & y_{0}^{\prime \prime} & y_{0}^{\prime \prime \prime} & \ldots & y_{0}^{(n+1)} \\
\cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot \\
y_{0}^{(n-1)} & y_{0}^{(n)} & y_{0}^{(n+1)} & \ldots & y_{0}^{(2 n-1)}
\end{array}\right|=0
Let us assume that the initial conditions (7) are chosen such that the characteristic equation (9^(')9^{\prime}) to have all the rootsr_(1),r_(2),dots,r_(n)r_{1}, r_{2}, \ldots, r_{n}distinct. In this case
{:(10)y=C_(1)e^(r_(1)x)+C_(2)e^(r_(2)x)+dots+C_(n)e^(r_(n)x):}\begin{equation*}
y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}+\ldots+C_{n} e^{r_{n} x} \tag{10}
\end{equation*}
is an integral of the differential equation (8) and will be an integral of the differential equation (1), if we show thatDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0, what happens when
{:(11)C_(1)C_(2)dotsC_(n)!=0:}\begin{equation*}
C_{1} C_{2} \ldots C_{n} \neq 0 \tag{11}
\end{equation*}
Indeed
Delta_(n-1)[y]=|[C_(1)e^(r_(1)x),C_(2)e^(r_(2)x),dots,C_(n)e^(r_(n)x)],[C_(1)r_(1)e^(r_(1)x),C_(2)r_(2)e^(r_(2)x),dots,C_(n)r_(n)e^(r_(n)x)],[*,*,*,*],[C_(1)r_(1)^(n-1)e^(r_(1)x),C_(2)r_(2)^(n-1)e^(r_(2)x),dots,C_(n)r_(n)^(n-1)e^(r_(n)x)]|*|[1,r_(1),dots,r_(1)^(n-1)],[1,r_(2),dots,r_(2)^(n-1)],[*,*,*,*],[1,r_(n),dots,r_(n)^(n-1)]|\Delta_{n-1}[y]=\left|\begin{array}{cccc}
C_{1} e^{r_{1} x} & C_{2} e^{r_{2} x} & \ldots & C_{n} e^{r_{n} x} \\
C_{1} r_{1} e^{r_{1} x} & C_{2} r_{2} e^{r_{2} x} & \ldots & C_{n} r_{n} e^{r_{n} x} \\
\cdot & \cdot & \cdot & \cdot \\
C_{1} r_{1}^{n-1} e^{r_{1} x} & C_{2} r_{2}^{n-1} e^{r_{2} x} & \ldots & C_{n} r_{n}^{n-1} e^{r_{n} x}
\end{array}\right| \cdot\left|\begin{array}{cccc}
1 & r_{1} & \ldots & r_{1}^{n-1} \\
1 & r_{2} & \ldots & r_{2}^{n-1} \\
\cdot & \cdot & \cdot & \cdot \\
1 & r_{n} & \ldots & r_{n}^{n-1}
\end{array}\right|
MEAN
{:(12)Delta_(n-1)[y]=C_(1)C_(2)dotsC_(n)V^(2)(r_(1),r_(2),dots,r_(n))e^((r_(1)+r_(2)+dots+r_(n))x):}\begin{equation*}
\Delta_{n-1}[y]=C_{1} C_{2} \ldots C_{n} V^{2}\left(r_{1}, r_{2}, \ldots, r_{n}\right) e^{\left(r_{1}+r_{2}+\ldots+r_{n}\right) x} \tag{12}
\end{equation*}
whereV(r_(1),r_(2),dots,r_(n))V\left(r_{1}, r_{2}, \ldots, r_{n}\right)is the Vandermonde determinant of distinct numbersr_(1),r_(2),dots,r_(n)r_{1}, r_{2}, \ldots, r_{n}. When condition (11) is satisfied, formula (10) is an integral of the differential equation (1). This integral is defined in the interval(-oo,+oo)(-\infty,+\infty)and the conditionDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0is valid in the interval(-oo,+oo)(-\infty,+\infty).
The integral that corresponds to Cauchy's conditions (7) is obtained by determiningA_(1),A_(2),dots,A_(n)A_{1}, A_{2}, \ldots, A_{n}from equations (6) and then putting
{:(13)C_(1)e^(r_(1)x_(0))=C_(1)^(')","quadC_(1)e^(r_(2)x_(0))=C_(2)^(')","dots","C_(n)e^(r_(n)x_(0))=C_(n)^('):}\begin{equation*}
C_{1} e^{r_{1} x_{0}}=C_{1}^{\prime}, \quad C_{1} e^{r_{2} x_{0}}=C_{2}^{\prime}, \ldots, C_{n} e^{r_{n} x_{0}}=C_{n}^{\prime} \tag{13}
\end{equation*}
it remains to be determinedC_(1)^('),C_(2)^('),dots,C_(n)^(')C_{1}^{\prime}, C_{2}^{\prime}, \ldots, C_{n}^{\prime}from the system of equations
{:[C_(1)^(')+C_(2)^(')+dots+C_(n)^(')=y_(0)],[(14)C_(1)^(')v_(1)+C_(2)^(')v_(2)+dots+C_(n)^(')v_(n)=y_(0)^(')],[C_(1)^(')v_(1)^(n-1)+C_(2)^(')v_(2)^(n-1)+dots+C_(n)^(')v_(n)^(n-1)=y_(0)^((n-1))]:}\begin{align*}
& C_{1}^{\prime}+C_{2}^{\prime}+\ldots+C_{n}^{\prime}=y_{0} \\
& C_{1}^{\prime} v_{1}+C_{2}^{\prime} v_{2}+\ldots+C_{n}^{\prime} v_{n}=y_{0}^{\prime} \tag{14}\\
& C_{1}^{\prime} v_{1}^{n-1}+C_{2}^{\prime} v_{2}^{n-1}+\ldots+C_{n}^{\prime} v_{n}^{n-1}=y_{0}^{(n-1)}
\end{align*}
We note that it is not possible for all numbersC_(1)^('),dots,C_(n)^(')C_{1}^{\prime}, \ldots, C_{n}^{\prime}to be null. It would follow thaty_(0)=y_(0)^(')=dots=y_(0)^((n-1))=0y_{0}=y_{0}^{\prime}=\ldots=y_{0}^{(n-1)}=0and so asDelta_(n-1)[y(x_(0))]=0\Delta_{n-1}\left[y\left(x_{0}\right)\right]=0, which is contrary to the hypothesis. Let us prove that it is not possible for even one ofC_(1)^('),dots,C_(n)^(')C_{1}^{\prime}, \ldots, C_{n}^{\prime}to be null.
Indeed, suppose we hadC_(1)^(')=0C_{1}^{\prime}=0Then the numbersr_(2),dots,r_(n)r_{2}, \ldots, r_{n}being distinct, between the equations
{:[C_(2)^(')+C_(3)^(')+dots+C_(n)^(')=y_(0)],[C_(2)^(')r_(2)+C_(3)^(')r_(3)+dots+C_(n)^(')r_(n)=y_(0)^(')],[C_(2)^(')r_(2)^(n-1)+C_(3)^(')r_(3)^(n-1)+dots+C_(n)^(')r_(n)^(n-1)=y_(0)^((n-1))]:}\begin{aligned}
& C_{2}^{\prime}+C_{3}^{\prime}+\ldots+C_{n}^{\prime}=y_{0} \\
& C_{2}^{\prime} r_{2}+C_{3}^{\prime} r_{3}+\ldots+C_{n}^{\prime} r_{n}=y_{0}^{\prime} \\
& C_{2}^{\prime} r_{2}^{n-1}+C_{3}^{\prime} r_{3}^{n-1}+\ldots+C_{n}^{\prime} r_{n}^{n-1}=y_{0}^{(n-1)}
\end{aligned}
can be removedC_(2)^('),dots,C_(n)^(')C_{2}^{\prime}, \ldots, C_{n}^{\prime}. We obtain a relation of the form
{:(15)y_(0)^((n-1))+p_(1)y_(0)^((n-2))+p_(2)y_(0)^((n-3))+dots+p_(n-1)y_(0)=0:}\begin{equation*}
y_{0}^{(n-1)}+p_{1} y_{0}^{(n-2)}+p_{2} y_{0}^{(n-3)}+\ldots+p_{n-1} y_{0}=0 \tag{15}
\end{equation*}
where
p_(1)=-(r_(2)+r_(3)+dots+r_(n)),quadp_(2)=r_(2)r_(3)+dots+r_(n-1)r_(n),dotsp_(n-1)=(-1)^(n-1)r_(2)r_(3)dotsr_(n)p_{1}=-\left(r_{2}+r_{3}+\ldots+r_{n}\right), \quad p_{2}=r_{2} r_{3}+\ldots+r_{n-1} r_{n}, \ldots p_{n-1}=(-1)^{n-1} r_{2} r_{3} \ldots r_{n}
In equations (6) let us replace
A_(1)=p_(1)-r_(1),A_(2)=p_(2)-p_(1)r_(1),A_(3)=p_(3)-p_(2)r_(1),dots,A_(n-1)=p_(n-1)-p_(n-2)r_(1)A_(n)=-p_(n-1)r_(1)A_{1}=p_{1}-r_{1}, A_{2}=p_{2}-p_{1} r_{1}, A_{3}=p_{3}-p_{2} r_{1}, \ldots, A_{n-1}=p_{n-1}-p_{n-2} r_{1} A_{n}=-p_{n-1} r_{1}The first
equation (6) is written
y_(0)^((n))+(p_(1)-r_(1))y_(0)^((n-1))+(p_(2)-p_(1)r_(1))y_(0)^((n-2))+dots+(p_(n-1)-p_(n-2)r_(1))y_(0)^(')-p_(n-1)r_(1)y_(0)=0y_{0}^{(n)}+\left(p_{1}-r_{1}\right) y_{0}^{(n-1)}+\left(p_{2}-p_{1} r_{1}\right) y_{0}^{(n-2)}+\ldots+\left(p_{n-1}-p_{n-2} r_{1}\right) y_{0}^{\prime}-p_{n-1} r_{1} y_{0}=0
and taking into account equation (15) the coefficient ofr_(1)r_{1}is zero, and the equation reduces to
p_(1)=-(r_(2)+r_(3)+dots+r_(n)),quadp_(2)=r_(2)r_(3)+dots+r_(n-1)r_(n),dotsp_(n-1)=(-1)^(n-1)r_(2)r_(3)dotsr_(n)p_{1}=-\left(r_{2}+r_{3}+\ldots+r_{n}\right), \quad p_{2}=r_{2} r_{3}+\ldots+r_{n-1} r_{n}, \ldots p_{n-1}=(-1)^{n-1} r_{2} r_{3} \ldots r_{n}
In equations (6) let us replace
A_(1)=p_(1)-r_(1),A_(2)=p_(2)-p_(1)r_(1),A_(3)=p_(3)-p_(2)r_(1),dots,A_(n-1)=p_(n-1)-p_(n-2)r_(1)A_(n)=-p_(n-1)r_(1)A_{1}=p_{1}-r_{1}, A_{2}=p_{2}-p_{1} r_{1}, A_{3}=p_{3}-p_{2} r_{1}, \ldots, A_{n-1}=p_{n-1}-p_{n-2} r_{1} A_{n}=-p_{n-1} r_{1}The first
equation (6) is written
y_(0)^((n))+(p_(1)-r_(1))y_(0)^((n-1))+(p_(2)-p_(1)r_(1))y_(0)^((n-2))+dots+(p_(n-1)-p_(n-2)r_(1))y_(0)^(')-p_(n-1)r_(1)y_(0)=0y_{0}^{(n)}+\left(p_{1}-r_{1}\right) y_{0}^{(n-1)}+\left(p_{2}-p_{1} r_{1}\right) y_{0}^{(n-2)}+\ldots+\left(p_{n-1}-p_{n-2} r_{1}\right) y_{0}^{\prime}-p_{n-1} r_{1} y_{0}=0
and taking into account equation (15) the coefficient ofr_(1)r_{1}is zero, and the equation reduces to
{:('")"y_(0)^((n))+p_(1)y_(0)^((n-1))+p_(2)y_(0)^((n-2))+dots+p_(n-1)y_(0)^(')=0.:}\begin{equation*}
y_{0}^{(n)}+p_{1} y_{0}^{(n-1)}+p_{2} y_{0}^{(n-2)}+\ldots+p_{n-1} y_{0}^{\prime}=0 . \tag{$\prime$}
\end{equation*}
Further, taking this into account, the second equation (6) reduces to
{:(('')")"y_(0)^((n+1))+p_(1)y_(0)^((n))+p_(2)y_(0)^((n-1))+dots+p_(n-1)y_(0)^('')=0:}\begin{equation*}
y_{0}^{(n+1)}+p_{1} y_{0}^{(n)}+p_{2} y_{0}^{(n-1)}+\ldots+p_{n-1} y_{0}^{\prime \prime}=0 \tag{$\prime\prime$}
\end{equation*}
and so on, until the penultimate equation is obtained
{:((''')")"y_(0)^((2n-2))+p_(1)y_(0)^((2n-3))+p_(2)y_(1)^((2n-4))+dots+p_(n-1)y_(1)^((n-1))=0:}\begin{equation*}
y_{0}^{(2 n-2)}+p_{1} y_{0}^{(2 n-3)}+p_{2} y_{1}^{(2 n-4)}+\ldots+p_{n-1} y_{1}^{(n-1)}=0 \tag{$\prime\prime\prime$}
\end{equation*}
But in this case the determinant
Delta_(n-1)[y(x_(0))]=|[y_(0),y_(0)^('),dots,y_(0)^((n-1))],[y_(0)^('),y_(0)^(''),dots,y_(0)^((n))],[*,*,*,*],[y_(0)^((n-1)),y_(0)^((n)),dots,y_(0)^((2n-2))]|\Delta_{n-1}\left[y\left(x_{0}\right)\right]=\left|\begin{array}{llll}
y_{0} & y_{0}^{\prime} & \ldots & y_{0}^{(n-1)} \\
y_{0}^{\prime} & y_{0}^{\prime \prime} & \ldots & y_{0}^{(n)} \\
\cdot & \cdot & \cdot & \cdot \\
y_{0}^{(n-1)} & y_{0}^{(n)} & \ldots & y_{0}^{(2 n-2)}
\end{array}\right|
is null because between the elements of the lines there is the same linear combination expressed by formulas (15),(15^(')),(15^('')),(15^('''))\left(15^{\prime}\right),\left(15^{\prime \prime}\right),\left(15^{\prime \prime \prime}\right)This, however, contradicts the hypothesis that
Delta_(n-1)[y(x)]!=0.\Delta_{n-1}[y(x)] \neq 0 .
So under the only condition that the characteristic equation (9^(')9^{\prime}) to have distinct roots, formula (10) in whichC_(1)C_(2)dotsC_(n)!=0C_{1} C_{2} \ldots C_{n} \neq 0is the integral of the differential equation (1).
Observation. Determination of constantsC_(1),C_(2),dots,C_(n);r_(1),r_(2),dots,r_(n)C_{1}, C_{2}, \ldots, C_{n} ; r_{1}, r_{2}, \ldots, r_{n}from formula (10), so thaty(x)y(x)to verify Cauchy's conditions (7), is done by putting
C_(1)e^(r_(1)x_(0))=C_(1)^('),quadC_(2)e^(r_(2)x_(0))=C_(2)^('),dots,C_(n)e^(r_(n)x_(0))=C_(n)^(')C_{1} e^{r_{1} x_{0}}=C_{1}^{\prime}, \quad C_{2} e^{r_{2} x_{0}}=C_{2}^{\prime}, \ldots, C_{n} e^{r_{n} x_{0}}=C_{n}^{\prime}
and solving the system of equations
{:[C_(1)^(')+C_(2)^(')+dots+C_(n)^(')=y_(0)],[(16)C_(1)^(')r_(1)+C_(2)^(')r_(2)+dots+C_(n)^(')r_(n)=y_(0)^(')],[C_(1)^(')r_(1)^(2n-1)+C_(2)^(')r_(2)^(2n-2)+dots+C_(n)^(')r_(n)^(2n-1)=y_(0)^((2n-1))]:}\begin{align*}
& C_{1}^{\prime}+C_{2}^{\prime}+\ldots+C_{n}^{\prime}=y_{0} \\
& C_{1}^{\prime} r_{1}+C_{2}^{\prime} r_{2}+\ldots+C_{n}^{\prime} r_{n}=y_{0}^{\prime} \tag{16}\\
& C_{1}^{\prime} r_{1}^{2 n-1}+C_{2}^{\prime} r_{2}^{2 n-2}+\ldots+C_{n}^{\prime} r_{n}^{2 n-1}=y_{0}^{(2 n-1)}
\end{align*}
in relationcuC_(1)^('),C_(2)^('),dots,C_(n)^(')\mathrm{cu} C_{1}^{\prime}, C_{2}^{\prime}, \ldots, C_{n}^{\prime}andr_(1),r_(2),dots,r_(n)r_{1}, r_{2}, \ldots, r_{n}.
His eliminationC_(1)^('),C_(2)^('),dots,C_(n)^(')C_{1}^{\prime}, C_{2}^{\prime}, \ldots, C_{n}^{\prime}between these equations leads to equations (6), where
His eliminationC_(1)^('),C_(2)^('),dots,C_(n)^(')C_{1}^{\prime}, C_{2}^{\prime}, \ldots, C_{n}^{\prime}between these equations leads to equations (6), where
A_(1)=-Sigmar_(1),A_(2)=Sigmar_(1)r_(2)dots,A_(n)=(-1)^(n)r_(1)r_(2)dotsr_(n)A_{1}=-\Sigma r_{1}, A_{2}=\Sigma r_{1} r_{2} \ldots, A_{n}=(-1)^{n} r_{1} r_{2} \ldots r_{n}
The equation that determinesr_(1),r_(2),dots,r_(n)r_{1}, r_{2}, \ldots, r_{n}is equation (9').
Example. Given the function
Example. Given the function
Y(x)=int_(alpha)^(beta)p(s)e^(sx)dsY(x)=\int_{\alpha}^{\beta} p(s) e^{s x} d s
wherep(s)p(s)is a positive function in the interval (alpha,beta\alpha, \beta), which can be cancelled inalpha\alphaandbeta\beta, one can determine an integral of the differential equation (1) that satisfies Cauchy's conditions.
y(x_(0))=Y(x_(0)),y^(')(x_(0))=Y^(')(x_(0)),dots,y^((2n-1))(x_(0))=Y^((2n-1))(x_(0))y\left(x_{0}\right)=Y\left(x_{0}\right), y^{\prime}\left(x_{0}\right)=Y^{\prime}\left(x_{0}\right), \ldots, y^{(2 n-1)}\left(x_{0}\right)=Y^{(2 n-1)}\left(x_{0}\right)
Indeed, the system of equations (16) corresponding to this case is
{:[C_(1)^(')+C_(2)^(')+dots+C_(n)^(')=int_(alpha)^(beta)p(s)e^(sx_(0))ds],[C_(1)^(')r_(1)+C_(2)^(')r_(2)+dots+C_(n)^(')r_(n)=int_(alpha)^(beta)sp(s)e^(sx_(0))ds],[C_(1)^(')r_(1)^(2n-1)+C_(2)^(')r_(2)^(2n-1)+dots+C_(n)^(')r_(n)^(2n-1)=int_(alpha)^(beta)s^(2n-1)p(s)e^(sx_(0))ds]:}\begin{aligned}
& C_{1}^{\prime}+C_{2}^{\prime}+\ldots+C_{n}^{\prime}=\int_{\alpha}^{\beta} p(s) e^{s x_{0}} d s \\
& C_{1}^{\prime} r_{1}+C_{2}^{\prime} r_{2}+\ldots+C_{n}^{\prime} r_{n}=\int_{\alpha}^{\beta} s p(s) e^{s x_{0}} d s \\
& C_{1}^{\prime} r_{1}^{2 n-1}+C_{2}^{\prime} r_{2}^{2 n-1}+\ldots+C_{n}^{\prime} r_{n}^{2 n-1}=\int_{\alpha}^{\beta} s^{2 n-1} p(s) e^{s x_{0}} d s
\end{aligned}
This system is classical; it is found in the theory of quadrature formulas. It is shown [5] that the numbersr_(1),r_(2),dots,r_(n)r_{1}, r_{2}, \ldots, r_{n}are all real, distinct, and contained withinalpha\alphaandbeta\beta, and the numbersC_(1)^('),C_(2)^('),dots,C_(n)^(')C_{1}^{\prime}, C_{2}^{\prime}, \ldots, C_{n}^{\prime}, that isC_(1),C_(2),dots,C_(n)C_{1}, C_{2}, \ldots, C_{n}are all positive.
3. The integrals of the differential equation (1) depend on the initial conditions (7). They can be such that the characteristic equation (9^(')9^{\prime}) of the differential equation (8) to have all distinct roots, and this case was studied in the previous point. However, it is possible that the initial conditions (9) are such that the characteristic equation (9^(')9^{\prime}) to have multiple roots.
3. The integrals of the differential equation (1) depend on the initial conditions (7). They can be such that the characteristic equation (9^(')9^{\prime}) of the differential equation (8) to have all distinct roots, and this case was studied in the previous point. However, it is possible that the initial conditions (9) are such that the characteristic equation (9^(')9^{\prime}) to have multiple roots.
Let us assume that the initial conditions (7) are such that the characteristic equation (9^(')9^{\prime}) has the rootr_(1)r_{1}multiple of the orderp_(1),r_(2)p_{1}, r_{2}, multiple of the orderp_(2),dots,r_(k)p_{2}, \ldots, r_{k}multiple of the orderp_(k)p_{k}where
p_(1)+p_(2)+dots+p_(k)=n.p_{1}+p_{2}+\ldots+p_{k}=n .
In this case the differential equation (8) has the integral
{:[y=((C_(1))/((p_(1)-1)!)x^(p_(1)-1)+(C_(11))/((p_(1)-2)!)x^(p_(1)-2)+dots+C_(1,p_(1)-1))e^(r_(1)x)],[(17)+((C_(2))/((p_(2)-1)!)x^(p_(2)-2)+(C_(21))/((p_(2)-2)!)x^(p_(2)-2)+dots+C_(2,p_(2)-1))e^(r_(2)x)],[+((C_(k))/((p_(k)-1)!)x^(p_(k)-1)+(C_(k1))/((p_(k)-2)!)x^(p_(k)-2)+dots+C_(k,p_(k)-1))e^(r_(k)x)]:}\begin{align*}
y & =\left(\frac{C_{1}}{\left(p_{1}-1\right)!} x^{p_{1}-1}+\frac{C_{11}}{\left(p_{1}-2\right)!} x^{p_{1}-2}+\ldots+C_{1, p_{1}-1}\right) e^{r_{1} x} \\
& +\left(\frac{C_{2}}{\left(p_{2}-1\right)!} x^{p_{2}-2}+\frac{C_{21}}{\left(p_{2}-2\right)!} x^{p_{2}-2}+\ldots+C_{2, p_{2}-1}\right) e^{r_{2} x} \tag{17}\\
& +\left(\frac{C_{k}}{\left(p_{k}-1\right)!} x^{p_{k}-1}+\frac{C_{k 1}}{\left(p_{k}-2\right)!} x^{p_{k}-2}+\ldots+C_{k, p_{k}-1}\right) e^{r_{k} x}
\end{align*}
and we have
Delta_(n-1)[y]=(-1)^(k)C_(1)^(p_(1))C_(2)^(p_(2))dotsC_(k)^(p_(k))V^(2)(ubrace(r_(1),dots,r_(1))_(p_(1)" ori ")ubrace(r_(2),dots,r_(2))_(p_(2)" ori "),quad,ubrace(r_(k),dots,r_(k))_(p_(k)" ori "))e^((p_(1)r_(1)+dots+p_(k)r_(k)x:})\Delta_{n-1}[y]=(-1)^{k} C_{1}^{p_{1}} C_{2}^{p_{2}} \ldots C_{k}^{p_{k}} V^{2}(\underbrace{r_{1}, \ldots, r_{1}}_{p_{1} \text { ori }} \underbrace{r_{2}, \ldots, r_{2}}_{p_{2} \text { ori }}, \quad, \underbrace{r_{k}, \ldots, r_{k}}_{p_{k} \text { ori }}) e^{\left(p_{1} r_{1}+\ldots+p_{k} r_{k} x\right.}
where the determinantV(r_(1),dots,r_(1);ubrace(r_(2),dots,r_(2))_(p_(1)" ori "),dots,ubrace(r^(k),dots,r_(k))_(p_(k)" ori "))\mathrm{V}(r_{1}, \ldots, r_{1} ; \underbrace{r_{2}, \ldots, r_{2}}_{p_{1} \text { ori }}, \ldots, \underbrace{r^{k}, \ldots, r_{k}}_{p_{k} \text { ori }})has the first column consisting of1,r_(1),dots,r_(1)^(n-1)1, r_{1}, \ldots, r_{1}^{n-1}, the second of the derivatives of these elements with respect tor_(1)r_{1}divided by 1!, the third with the second-order derivatives divided by2!,dots,ak2!, \ldots, a k-a of the derivatives of the orderkk-1 of the elements of the first column divided by (k-1k-1)!. The followingk_(2)k_{2}columns are formed in the same way, replacingr_(1)r_{1}withr_(2)dotsr_{2} \ldots, and so on.
It follows that ifC_(1)C_(2)dotsC_(k)!=0,yC_{1} C_{2} \ldots C_{k} \neq 0, yis the integral of the differential equation (1). And this integral is defined in the interval (-oo,+oo-\infty,+\infty), and the conditionDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0is also valid in the interval (-oo,+oo-\infty,+\infty).
Formula (18) can be proven directly, but the calculations are complicated.
However, it is known that the integrals of a differential equation with constant coefficients corresponding to a multiple root can be obtained from the integrals corresponding to distinct roots, by a passage to the limit. We will use this passage to the limit to prove formula (8).
However, it is known that the integrals of a differential equation with constant coefficients corresponding to a multiple root can be obtained from the integrals corresponding to distinct roots, by a passage to the limit. We will use this passage to the limit to prove formula (8).
Suppose we are dealing with the case when the rootr_(1)r_{1}is double, the other roots being distinct.
Puttingr_(2)=r_(1)+hr_{2}=r_{1}+h, forh!=0h \neq 0, the differential equation (8) has the integral
{:(19)y=C_(1)(e^((r_(1)+h)x)-e^(r_(1)x))/(h)+C_(2)e^(r_(1)x)+C_(3)e^(r_(3)x)+dots+C_(n)e^(r_(n)x):}\begin{equation*}
y=C_{1} \frac{e^{\left(r_{1}+h\right) x}-e^{r_{1} x}}{h}+C_{2} e^{r_{1} x}+C_{3} e^{r_{3} x}+\ldots+C_{n} e^{r_{n} x} \tag{19}
\end{equation*}
which we can write in the form
{:(20)y=C_(1)^(')e^(r_(1)x)+C_(2)^(')e^((r_(1)+h]x)+C_(3)e^(r_(3)x)+dots+C_(n)e^(r_(n)x):}\begin{equation*}
y=C_{1}^{\prime} e^{r_{1} x}+C_{2}^{\prime} e^{\left(r_{1}+h\right] x}+C_{3} e^{r_{3} x}+\ldots+C_{n} e^{r_{n} x} \tag{20}
\end{equation*}
where
{:(21)C_(1)^(')=C_(2)-(C_(1))/(h)","quadC_(2)^(')=(C_(1))/(h):}\begin{equation*}
C_{1}^{\prime}=C_{2}-\frac{C_{1}}{h}, \quad C_{2}^{\prime}=\frac{C_{1}}{h} \tag{21}
\end{equation*}
Whenh rarr0h \rightarrow 0, functionyyhas a limit
{:(22)y_(1)=(C_(1)x+C_(2))e^(r_(1)x)+C_(3)e^(r_(3)x)+dots+C_(n)er_(n)x:}\begin{equation*}
y_{1}=\left(C_{1} x+C_{2}\right) e^{r_{1} x}+C_{3} e^{r_{3} x}+\ldots+C_{n} e r_{n} x \tag{22}
\end{equation*}
andy_(1)y_{1}is the integral of the differential equation (8). It is easy to prove that the derivative of any orderppof the functionyy, tends to the derivative of the same order of the functiony_(1)y_{1}whenh rarr0h \rightarrow 0, from which it follows that
{:(23)lim_(h rarr0)Delta_(n rarr1)[y]=Delta_(n-1)[y_(1)].:}\begin{equation*}
\lim _{h \rightarrow 0} \Delta_{n \rightarrow 1}[y]=\Delta_{n-1}\left[y_{1}\right] . \tag{23}
\end{equation*}
But applying formula (12) we have
Delta_(n-1)[y]=C_(1)^(')C_(2)^(')C_(3)dotsC_(n)V^(2)(r_(1),r_(1)+h,r_(3),dots,r_(n))e^((2r_(1)+h+r_(3)+dots+r_(n))x)\Delta_{n-1}[y]=C_{1}^{\prime} C_{2}^{\prime} C_{3} \ldots C_{n} V^{2}\left(r_{1}, r_{1}+h, r_{3}, \ldots, r_{n}\right) e^{\left(2 r_{1}+h+r_{3}+\ldots+r_{n}\right) x}
ReplacingC_(1)^('),C_(2)^(')cuC_{1}^{\prime}, C_{2}^{\prime} \mathrm{cu}formulas (21), we will have
Delta_(n-1)[y]=C_(1)(C_(2)h-C_(1))C_(3)dotsC_(n)[(V(r_(1),r_(1)+h,r_(3),dots,r_(n)))/(h)]^(2)e^((2r_(1)+h+r_(3)+dots+r_(n))x)\Delta_{n-1}[y]=C_{1}\left(C_{2} h-C_{1}\right) C_{3} \ldots C_{n}\left[\frac{V\left(r_{1}, r_{1}+h, r_{3}, \ldots, r_{n}\right)}{h}\right]^{2} e^{\left(2 r_{1}+h+r_{3}+\ldots+r_{n}\right) x}
When we doh rarr0h \rightarrow 0, we obtain
lim_(h rarr0)Delta_(n-1)[y]=-C_(1)^(2)C_(3)dotsC_(n)V^(2)(r_(1),r_(1),r_(3),dots,r_(n))e^((2r_(1)+r_(3)+dots+r_(n))x)\lim _{h \rightarrow 0} \Delta_{n-1}[y]=-C_{1}^{2} C_{3} \ldots C_{n} V^{2}\left(r_{1}, r_{1}, r_{3}, \ldots, r_{n}\right) e^{\left(2 r_{1}+r_{3}+\ldots+r_{n}\right) x}
because
lim_(h rarr0)(1)/(h)|[1,1,1,dots,1],[r_(1),r_(1)+h,r_(3),dots,r_(n)],[r_(1)^(2),(r_(1)+h)^(2),r_(3)^(2),dots,r_(n)^(2)],[r_(1)^(3),(r_(1)+h)^(3),r_(3)^(3),dots,r_(n)^(3)],[cdots,cdots,cdots,cdots,cdots],[r_(1)^(n-1),(r_(1)+h)^(n-1),r_(3)^(n-1),dots,r_(n)^(n-1)]|=V(r_(1),r_(2),r_(3)dots,r_(n)).\lim _{h \rightarrow 0} \frac{1}{h}\left|\begin{array}{ccccc}
1 & 1 & 1 & \ldots & 1 \\
r_{1} & r_{1}+h & r_{3} & \ldots & r_{n} \\
r_{1}^{2} & \left(r_{1}+h\right)^{2} & r_{3}^{2} & \ldots & r_{n}^{2} \\
r_{1}^{3} & \left(r_{1}+h\right)^{3} & r_{3}^{3} & \ldots & r_{n}^{3} \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
r_{1}^{n-1} & \left(r_{1}+h\right)^{n-1} & r_{3}^{n-1} & \ldots & r_{n}^{n-1}
\end{array}\right|=V\left(r_{1}, r_{2}, r_{3} \ldots, r_{n}\right) .
Taking into account formula (23), we thus deduce that
{:(24)Delta_(n-1)[y_(1)]=-C_(1)^(2)C_(3)dotsC_(n)V^(2)(r_(1),r_(1),r_(3),dots,r_(n))e^((2r_(1)+r_(3)+dots+r_(n))x):}\begin{equation*}
\Delta_{n-1}\left[y_{1}\right]=-C_{1}^{2} C_{3} \ldots C_{n} V^{2}\left(r_{1}, r_{1}, r_{3}, \ldots, r_{n}\right) e^{\left(2 r_{1}+r_{3}+\ldots+r_{n}\right) x} \tag{24}
\end{equation*}
Let us now suppose that we are dealing with the case when the rootr_(1)r_{1}is triple, the others being distinct.
In the integral (22) of the differential equation (8), let us putr_(3)=r_(1)+hr_{3}=r_{1}+hand let's write this integral in the form
y=C_(1)(e^((r_(1)+h)x)-e^(r_(1)x)-hxe^(r_(1)x))/(h^(2))+C_(2)(e^((r_(1)+h)x)-e^(r_(1)x))/(h)+C_(3)e^(r_(1)x)+C_(4)e^(r_(4)x)+dots+C_(n)e^(r_(n)x)y=C_{1} \frac{e^{\left(r_{1}+h\right) x}-e^{r_{1} x}-h x e^{r_{1} x}}{h^{2}}+C_{2} \frac{e^{\left(r_{1}+h\right) x}-e^{r_{1} x}}{h}+C_{3} e^{r_{1} x}+C_{4} e^{r_{4} x}+\ldots+C_{n} e^{r_{n} x}
or
y=C_(1)(e^((r_(1)+h)x)-e^(r_(1)x)-hxe^(r_(1)x))/(h^(2))+C_(2)(e^((r_(1)+h)x)-e^(r_(1)x))/(h)+C_(3)e^(r_(1)x)+C_(4)e^(r_(4)x)+dots+C_(n)e^(r_(n)x)y=C_{1} \frac{e^{\left(r_{1}+h\right) x}-e^{r_{1} x}-h x e^{r_{1} x}}{h^{2}}+C_{2} \frac{e^{\left(r_{1}+h\right) x}-e^{r_{1} x}}{h}+C_{3} e^{r_{1} x}+C_{4} e^{r_{4} x}+\ldots+C_{n} e^{r_{n} x}
or
y=(C_(1)^(')x+C_(2)^('))^(r_(1)x)+C_(3)^(')e^((r_(1)+h)x)+C_(4)e^(r_(4)x)+dots+C_(n)e^(r_(n)x)y=\left(C_{1}^{\prime} x+C_{2}^{\prime}\right)^{r_{1} x}+C_{3}^{\prime} e^{\left(r_{1}+h\right) x}+C_{4} e^{r_{4} x}+\ldots+C_{n} e^{r_{n} x}
where
{:(25)C_(1)^(')=-(C_(1))/(h)","quadC_(2)^(')=-(C_(1))/(h^(2))-(C_(2))/(h)+C_(3)","quadC_(3)^(')=(C_(1))/(h^(2))+(C_(2))/(h):}\begin{equation*}
C_{1}^{\prime}=-\frac{C_{1}}{h}, \quad C_{2}^{\prime}=-\frac{C_{1}}{h^{2}}-\frac{C_{2}}{h}+C_{3}, \quad C_{3}^{\prime}=\frac{C_{1}}{h^{2}}+\frac{C_{2}}{h} \tag{25}
\end{equation*}
Whenh rarr0,yh \rightarrow 0, ytends towards the integral
{:(26)y_(2)=(C_(1)(x^(2))/(2!)+C_(2)(x)/(1!)+C_(3))e^(r_(1)x)+C_(4)e^(r_(4)x)+dots+C_(n)e^(r_(n)x):}\begin{equation*}
y_{2}=\left(C_{1} \frac{x^{2}}{2!}+C_{2} \frac{x}{1!}+C_{3}\right) e^{r_{1} x}+C_{4} e^{r_{4} x}+\ldots+C_{n} e^{r_{n} x} \tag{26}
\end{equation*}
of the differential equation (8) and it is shown that
{:(27)lim_(h rarr0)Delta_(n-1)[y]=Delta_(n-1)[y_(2)].:}\begin{equation*}
\lim _{h \rightarrow 0} \Delta_{n-1}[y]=\Delta_{n-1}\left[y_{2}\right] . \tag{27}
\end{equation*}
On the other hand, taking into account equation (24) and formulas (25), we can write
Delta_(n-1)[y]=-C_(1)^(2)(C_(1)+hC_(2))C_(4)dotsC_(n)[([V(r_(1),r_(1),r_(1)+h,r_(4),dots,r_(n))])/(h^(2))e^((3r_(1)+h+r_(4)+dots+r_(n))x):}\Delta_{n-1}[y]=-C_{1}^{2}\left(C_{1}+h C_{2}\right) C_{4} \ldots C_{n}\left[\frac{\left[V\left(r_{1}, r_{1}, r_{1}+h, r_{4}, \ldots, r_{n}\right)\right]}{h^{2}} e^{\left(3 r_{1}+h+r_{4}+\ldots+r_{n}\right) x}\right.
Taking into account that
lim_(h rarr0)(1)/(h^(2))|[1,0,1,1,dots,1],[r_(1),1,r_(1)+h,r_(4),dotsr_(n)],[r_(1)^(2),C_(2)^(1)r_(1),(r_(1)+h)^(2),r_(4)^(2),dotsr_(n)^(2)],[cdots,cdots,cdots,cdots,cdots,cdots],[r_(1)^(n-1),C_(n-1)^(1)r_(1)^(n-2),(r_(1)+h)^(n-1),r_(4)^(n-1),dotsr_(n)^(n-1)]|=V(r_(1),r_(1),r_(1),r_(4),dots,r_(n))\lim _{h \rightarrow 0} \frac{1}{h^{2}}\left|\begin{array}{cccccc}
1 & 0 & 1 & 1 & \ldots & 1 \\
r_{1} & 1 & r_{1}+h & r_{4} & \ldots r_{n} \\
r_{1}^{2} & C_{2}^{1} r_{1} & \left(r_{1}+h\right)^{2} & r_{4}^{2} & \ldots r_{n}^{2} \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
r_{1}^{n-1} & C_{n-1}^{1} r_{1}^{n-2} & \left(r_{1}+h\right)^{n-1} & r_{4}^{n-1} & \ldots r_{n}^{n-1}
\end{array}\right|=V\left(r_{1}, r_{1}, r_{1}, r_{4}, \ldots, r_{n}\right)
and taking the limit in the previous formula, we deduce that
lim_(h rarr0)Delta_(n rarr1)[y]=-C_(1)^(3)C_(4)dotsC_(n)V^(2)(r_(1),r_(1),r_(1)r_(4),dots,r_(n))e^((3r_(1)+r_(4)+dots+r_(n))x)\lim _{h \rightarrow 0} \Delta_{n \rightarrow 1}[y]=-\mathrm{C}_{1}^{3} C_{4} \ldots C_{n} V^{2}\left(r_{1}, r_{1}, r_{1} r_{4}, \ldots, r_{n}\right) e^{\left(3 r_{1}+r_{4}+\ldots+r_{n}\right) x}
and according to formula (27), we will have
Delta_(n-1)[y_(2)]=-C_(1)^(3)C_(4)dotsC_(n)V^(2)(r_(1),r_(1),r_(1),r_(4),dots,r_(n))e^((3r_(1)+r_(4)+dots+r_(n))x)\Delta_{n-1}\left[y_{2}\right]=-C_{1}^{3} C_{4} \ldots C_{n} V^{2}\left(r_{1}, r_{1}, r_{1}, r_{4}, \ldots, r_{n}\right) e^{\left(3 r_{1}+r_{4}+\ldots+r_{n}\right) x}
With this we consider that we have given sufficient clarifications to be able to finish the proof of formula (18).
4. In summary, the integral (10) of the differential equation (1) was highlighted in whichC_(1)C_(2)dotsC_(n)!=0C_{1} C_{2} \ldots C_{n} \neq 0, as well as integrals of the form (17), in whichr_(1),r_(2),dots,r_(k)r_{1}, r_{2}, \ldots, r_{k}are distinct numbers,kkbeing 1 , or2,dots2, \ldots, orn-1n-1, the polynomials that multiplye^(r_(1)x),e^(r_(2)x),dots,e^(r_(k)x)e^{r_{1} x}, e^{r_{2} x}, \ldots, e^{r_{k} x}being of effective degreesp_(1)-1,p_(2)-1,dots,p_(k-1)p_{1}-1, p_{2}-1, \ldots, p_{k-1}, wherep_(1)+p_(2)+dots+p_(k)=np_{1}+p_{2}+\ldots+p_{k}=n, that isC_(1)C_(2)dotsC_(k)!=0C_{1} C_{2} \ldots C_{k} \neq 0For all these integralsDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0.
4. In summary, the integral (10) of the differential equation (1) was highlighted in whichC_(1)C_(2)dotsC_(n)!=0C_{1} C_{2} \ldots C_{n} \neq 0, as well as integrals of the form (17), in whichr_(1),r_(2),dots,r_(k)r_{1}, r_{2}, \ldots, r_{k}are distinct numbers,kkbeing 1 , or2,dots2, \ldots, orn-1n-1, the polynomials that multiplye^(r_(1)x),e^(r_(2)x),dots,e^(r_(k)x)e^{r_{1} x}, e^{r_{2} x}, \ldots, e^{r_{k} x}being of effective degreesp_(1)-1,p_(2)-1,dots,p_(k-1)p_{1}-1, p_{2}-1, \ldots, p_{k-1}, wherep_(1)+p_(2)+dots+p_(k)=np_{1}+p_{2}+\ldots+p_{k}=n, that isC_(1)C_(2)dotsC_(k)!=0C_{1} C_{2} \ldots C_{k} \neq 0For all these integralsDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0.
Let us now prove that any integral of the equationDelta_(q)[y]=0\Delta_{q}[y]=0for whichDelta_(q-1)[y]!=0\Delta_{q-1}[y] \neq 0, whereq < nq<n, is the integral of the differential equationDelta_(n)[y]=0\Delta_{n}[y]=0.
Whether
Delta_(q)[y]=|[y,y^('),dots,y^((q-1)),y^((q))],[y^('),y^(''),dots,y^((q)),y^((q+1))],[*,*,*,*,*],[y^((q)),y^((q+1)),dots,y^((2q-1)),y^((q))]|=0\Delta_{q}[y]=\left|\begin{array}{ccccc}
y & y^{\prime} & \ldots & y^{(q-1)} & y^{(q)} \\
y^{\prime} & y^{\prime \prime} & \ldots & y^{(q)} & y^{(q+1)} \\
\cdot & \cdot & \cdot & \cdot & \cdot \\
y^{(q)} & y^{(q+1)} & \ldots & y^{(2 q-1)} & y^{(q)}
\end{array}\right|=0
the differential equation andyyan integral of it for whichDelta_(q-1)[y]!=0\Delta_{q-1}[y] \neq 0. It is proved as in no. 1, that between the elements of the columns of the determinantDelta_(q)[y]\Delta_{q}[y]there is the same linear relationship with constant coefficients, i.e.
{:[B_(q)y+B_(q-1)y^(')+dots+B_(1)y^((q-1))+y^((q))=0],[(28)B_(q)y^(')+B_(q-1)y^('')+dots+B_(1)y^((q))+y^((q+1))=0],[B_(q)y^((q))+B_(q-1)y^((q+1))+dots+B_(1)y^((2q-1))+y^((2q))=0.]:}\begin{align*}
& B_{q} y+B_{q-1} y^{\prime}+\ldots+B_{1} y^{(q-1)}+y^{(q)}=0 \\
& B_{q} y^{\prime}+B_{q-1} y^{\prime \prime}+\ldots+B_{1} y^{(q)}+y^{(q+1)}=0 \tag{28}\\
& B_{q} y^{(q)}+B_{q-1} y^{(q+1)}+\ldots+B_{1} y^{(2 q-1)}+y^{(2 q)}=0 .
\end{align*}
But the coefficientsB_(1),B_(2),dots,B_(q)B_{1}, B_{2}, \ldots, B_{q}being constants, we can successively derive the last equation and we will obtain
{:[('")"B_(q)y^((q+1))+B_(q-1)y^((q+2))+dots+B_(1)y^((2q))+y^((2q+1))=0],[B_(q)y^((n))+B_(q-1)y^((n+1))+dots+B_(1)y^((n+q-1))+y^((n+q))=0]:}\begin{align*}
& B_{q} y^{(q+1)}+B_{q-1} y^{(q+2)}+\ldots+B_{1} y^{(2 q)}+y^{(2 q+1)}=0 \tag{$\prime$}\\
& B_{q} y^{(n)}+B_{q-1} y^{(n+1)}+\ldots+B_{1} y^{(n+q-1)}+y^{(n+q)}=0
\end{align*}
We thus deduce that
Delta_(n)[y]=0\Delta_{n}[y]=0
because between the elements of the firstq+1q+1columns of there is the same linear relationship, as shown by formulas (28) and (28').
5. Conclusion. Let yo be the integral of the equationDelta_(n)[y]=0\Delta_{n}[y]=0If she doesn't check the equationDelta_(n-1)[y]=0\Delta_{n-1}[y]=0, has the form (10), whereC_(1)C_(2)dotsC_(n)!=0C_{1} C_{2} \ldots C_{n} \neq 0, or the form (17), wherek=1k=1, or2,dots2, \ldots, orn-1n-1, andp_(1),p_(2),p_(3),dots,p_(k)p_{1}, p_{2}, p_{3}, \ldots, p_{k}are positive or zero integers such thatp_(1)+p_(2)+dots+p_(k)=np_{1}+p_{2}+\ldots+p_{k}=n, andC_(1)C_(2)dotsC_(k)!=0C_{1} C_{2} \ldots C_{k} \neq 0.
5. Conclusion. Let yo be the integral of the equationDelta_(n)[y]=0\Delta_{n}[y]=0If she doesn't check the equationDelta_(n-1)[y]=0\Delta_{n-1}[y]=0, has the form (10), whereC_(1)C_(2)dotsC_(n)!=0C_{1} C_{2} \ldots C_{n} \neq 0, or the form (17), wherek=1k=1, or2,dots2, \ldots, orn-1n-1, andp_(1),p_(2),p_(3),dots,p_(k)p_{1}, p_{2}, p_{3}, \ldots, p_{k}are positive or zero integers such thatp_(1)+p_(2)+dots+p_(k)=np_{1}+p_{2}+\ldots+p_{k}=n, andC_(1)C_(2)dotsC_(k)!=0C_{1} C_{2} \ldots C_{k} \neq 0.
But if the integralyyof the equationDelta_(n)[y]=0\Delta_{n}[y]=0check the equationDelta_(n-1)[y]=0\Delta_{n-1}[y]=0but it doesn't check the equationDelta_(n-2)[y=0\Delta_{n-2}[y=0, it also has one of the forms shown in the previous paragraph, butnnchanges ton-1n-1.
In general, if the integralyyof the equationDelta_(n)[y]=0\Delta_{n}[y]=0check the equations
Delta_(n-1)[y]=0,quadDelta_(n-2)[y]=0,dots,Delta_(n-j)[y]=0\Delta_{n-1}[y]=0, \quad \Delta_{n-2}[y]=0, \ldots, \Delta_{n-j}[y]=0
but
Delta_(n rarr j-1)[y]!=0\Delta_{n \rightarrow j-1}[y] \neq 0
then it has one of the forms shown in paragraph 1, provided that it changesnninn-jn-j.
Example. Consider the differential equation
Delta_(3)[y]=0.\Delta_{3}[y]=0 .
1^(@)1^{\circ}. If an integral of ityyit is such thatDelta_(2)[y]!=0\Delta_{2}[y] \neq 0, then it has one of the following forms
y=C_(1)e^(r_(1)x)+C_(2)e^(r_(2)x)+C_(3)e^(r_(2)x)y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}+C_{3} e^{r_{2} x}
wheregamma_(1),gamma_(2),gamma_(3)\gamma_{1}, \gamma_{2}, \gamma_{3}are distinct numbers andC_(1)C_(2)C_(3)!=0C_{1} C_{2} C_{3} \neq 0, or
y=(C_(1)x+C_(2))e^(r_(1)x)+C_(3)e^(r_(3)x)y=\left(C_{1} x+C_{2}\right) e^{r_{1} x}+C_{3} e^{r_{3} x}
wheregamma_(1),gamma_(3)\gamma_{1}, \gamma_{3}are distinct numbers andC_(1)C_(2)!=0C_{1} C_{2} \neq 0or
y=(C_(1)(x^(2))/(2!)+C_(2)(x)/(1!)+C_(3))e^(r_(1)x)y=\left(C_{1} \frac{x^{2}}{2!}+C_{2} \frac{x}{1!}+C_{3}\right) e^{r_{1} x}
whereC_(1)!=0C_{1} \neq 0.
2^(@)2^{\circ}If the integralyyof the differential equationDelta_(3)[y]=0\Delta_{3}[y]=0check the equationDelta_(2)[y]=0\Delta_{2}[y]=0, butDelta_(1)[y]!=0\Delta_{1}[y] \neq 0, then it has one of the following forms
2^(@)2^{\circ}If the integralyyof the differential equationDelta_(3)[y]=0\Delta_{3}[y]=0check the equationDelta_(2)[y]=0\Delta_{2}[y]=0, butDelta_(1)[y]!=0\Delta_{1}[y] \neq 0, then it has one of the following forms
y=C_(1)e^(r_(1)x)+C_(2)e^(r_(2)x)y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}
wherer_(1)r_{1}andr_(2)r_{2}are distinct numbers andC_(1)C_(2)!=0C_{1} C_{2} \neq 0, or
y=(C_(1)x+C_(2))e^(r_(1)x)y=\left(C_{1} x+C_{2}\right) e^{r_{1} x}
whereC_(1)!=0C_{1} \neq 0.
3^(@)3^{\circ}If the integralyyof the equationDelta_(3)[y]=0\Delta_{3}[y]=0check the equationsDelta_(2)[y]=0\Delta_{2}[y]=0,Delta_(1)[y]=0\Delta_{1}[y]=0, butDelta_(0)[y]=y!=0\Delta_{0}[y]=y \neq 0, then it has the form
3^(@)3^{\circ}If the integralyyof the equationDelta_(3)[y]=0\Delta_{3}[y]=0check the equationsDelta_(2)[y]=0\Delta_{2}[y]=0,Delta_(1)[y]=0\Delta_{1}[y]=0, butDelta_(0)[y]=y!=0\Delta_{0}[y]=y \neq 0, then it has the form
y=C_(1)e^(r_(1)x)y=C_{1} e^{r_{1} x}
whereC_(1)!=0C_{1} \neq 0.
4^(@)4^{\circ}. If, finally, the integralyyof the differential equationDelta_(3)[y]=0\Delta_{3}[y]=0check the equationsDelta_(2)[y]=0,Delta_(1)[y]=0,Delta_(0)[y]=0\Delta_{2}[y]=0, \Delta_{1}[y]=0, \Delta_{0}[y]=0, it is obviously identically zero.
6. Let us proceed to the integration of the differential equation
4^(@)4^{\circ}. If, finally, the integralyyof the differential equationDelta_(3)[y]=0\Delta_{3}[y]=0check the equationsDelta_(2)[y]=0,Delta_(1)[y]=0,Delta_(0)[y]=0\Delta_{2}[y]=0, \Delta_{1}[y]=0, \Delta_{0}[y]=0, it is obviously identically zero.
6. Let us proceed to the integration of the differential equation
{:(29)Delta_(n)[y]=Ae^(alpha x):}\begin{equation*}
\Delta_{n}[y]=A e^{\alpha x} \tag{29}
\end{equation*}
whereAAandalpha\alphaare constant andA!=0A \neq 0For this
, we will use the identity
, we will use the identity
{:(30)|[Delta_(n)[y],Delta_(n)^(')[y]],[Delta_(n)^(')[y],Delta_(n)^('')[y]]|=Delta_(n-1)[y]Delta_(n+1)[y]:}\left|\begin{array}{ll}
\Delta_{n}[y] & \Delta_{n}^{\prime}[y] \tag{30}\\
\Delta_{n}^{\prime}[y] & \Delta_{n}^{\prime \prime}[y]
\end{array}\right|=\Delta_{n-1}[y] \Delta_{n+1}[y]
which is proven in the following way.
Let us write the determinantDelta_(n+1)[y]\Delta_{n+1}[y]in the form of
Let us write the determinantDelta_(n+1)[y]\Delta_{n+1}[y]in the form of
D=|[a_(11),dots,a_(1n),a_(1,n+1)],[vdots,,vdots,vdots],[vdots,dots,vdots,vdots],[a_(n1),dots,a_(nn),a_(n,n+1)],[a_(n+1,1),dots,a_(n+1,n),a_(n+1,n+1)]|D=\left|\begin{array}{cccc}
a_{11} & \ldots & a_{1 n} & a_{1, n+1} \\
\vdots & & \vdots & \vdots \\
\vdots & \ldots & \vdots & \vdots \\
a_{n 1} & \ldots & a_{n n} & a_{n, n+1} \\
a_{n+1,1} & \ldots & a_{n+1, n} & a_{n+1, n+1}
\end{array}\right|
and let's notice that we have
Delta_(n)[y]=(del D)/(dela_(n+1,n+1)),quadDelta_(n)^(')[y]=-(del D)/(dela_(n,n+1))=-(del D)/(dela_(r+1,n)),quadDelta_(n)^('')[y]=(del D)/(dela_(n,n)).\Delta_{n}[y]=\frac{\partial D}{\partial a_{n+1, n+1}}, \quad \Delta_{n}^{\prime}[y]=-\frac{\partial D}{\partial a_{n, n+1}}=-\frac{\partial D}{\partial a_{r+1, n}}, \quad \Delta_{n}^{\prime \prime}[y]=\frac{\partial D}{\partial a_{n, n}} .
It is known that we have the identity [4]
|[(del D)/(dela_(n,n)),(del D)/(dela_(n,n+1))],[(del D)/(dela_(n+1,n)),(del D)/(dela_(n+1,n+1))]|=D_(2)D_(1)\left|\begin{array}{cc}
\frac{\partial D}{\partial a_{n, n}} & \frac{\partial D}{\partial a_{n, n+1}} \\
\frac{\partial D}{\partial a_{n+1, n}} & \frac{\partial D}{\partial a_{n+1, n+1}}
\end{array}\right|=D_{2} D_{1}
whereD_(2)D_{2}is the determinant obtained fromDDby deleting the last two rows and columns. This identity leads to identity (30).
Whetheryyan integral of the differential equation (29). It is obvious thatu=Ae^(ax)u=A e^{a x}being an integral of the differential equationDelta_(1)[u]=0\Delta_{1}[u]=0, we will have applying identity (30)
Delta_(n-1)[y]Delta_(n+1)[y]=0.\Delta_{n-1}[y] \Delta_{n+1}[y]=0 .
But it can't be thatyyto check the equationDelta_(n-1)[y]=0\Delta_{n-1}[y]=0, because we would also haveDelta_(n)[y]=0\Delta_{n}[y]=0, which is impossible becauseA_(!=0)A_{\neq 0}. So any integral of the differential equation (29) is an integral of the differential equation
{:(31)Delta_(n+1)[y]=0:}\begin{equation*}
\Delta_{n+1}[y]=0 \tag{31}
\end{equation*}
and therefore the integrals of the differential equation (29) must be sought among the integrals of the differential equation (31).
Let us first consider the integral of the differential equation (31)
y=C_(1)e^(r_(1)x)+C_(2)e^(r_(2)x)+dots+C_(n)e^(r_(n)x)+C_(n+1)e^(r_(n+1)x)y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}+\ldots+C_{n} e^{r_{n} x}+C_{n+1} e^{r_{n+1} x}
for whichDelta_(n)[y]!=0\Delta_{n}[y] \neq 0, that isr_(1),r_(2),dots,r_(n),r_(n+1)r_{1}, r_{2}, \ldots, r_{n}, r_{n+1}are distinct numbers andC_(1)C_(2)dotsC_(n)C_(n+1)!=0C_{1} C_{2} \ldots C_{n} C_{n+1} \neq 0According to formula (12), we have
Delta_(n)[y]=C_(1)C_(2)dotsC_(n)^(-)C_(n+1)V^(2)(r_(1),r_(2),dots,r_(n),r_(n+1))e^((r_(1)+r_(2)+dots+r_(n)+r_(n+1))x)\Delta_{n}[y]=C_{1} C_{2} \ldots C_{n}^{-} C_{n+1} V^{2}\left(r_{1}, r_{2}, \ldots, r_{n}, r_{n+1}\right) e^{\left(r_{1}+r_{2}+\ldots+r_{n}+r_{n+1}\right) x}
For equation (29) to be satisfied, we will choose
{:[r_(1)+r_(2)+dots+r_(n)+r_(n+1)=alpha],[C_(1)C_(2)dotsC_(n)C_(n+1)V^(2)(r_(1),r_(2),dots,r_(n),r_(n+1))=A]:}\begin{aligned}
& r_{1}+r_{2}+\ldots+r_{n}+r_{n+1}=\alpha \\
& C_{1} C_{2} \ldots C_{n} C_{n+1} V^{2}\left(r_{1}, r_{2}, \ldots, r_{n}, r_{n+1}\right)=A
\end{aligned}
The first equation determines ther_(n+1)r_{n+1}, that is
r_(n+1)=alpha-(r_(1)+r_(2)+dots+r_(n))r_{n+1}=\alpha-\left(r_{1}+r_{2}+\ldots+r_{n}\right)
and we will chooser_(1),r_(2),dots,r_(n)r_{1}, r_{2}, \ldots, r_{n}so thatr_(1),r_(2),dots,r_(n),r_(n+1)r_{1}, r_{2}, \ldots, r_{n}, r_{n+1}to be distinct numbers. Then the second equation determinesC_(n+1)C_{n+1}and we will have
C_(n+1)=(A)/(C_(1)C_(2)dotsC_(n)V^(2)[r_(1),r_(2)dotsr_(n),alpha-(r_(1)+r_(2)+dots+r_(n))])C_{n+1}=\frac{A}{C_{1} C_{2} \ldots C_{n} V^{2}\left[r_{1}, r_{2} \ldots r_{n}, \alpha-\left(r_{1}+r_{2}+\ldots+r_{n}\right)\right]}
The integral of equation (29) is presented in the form
y=C_(1)e^(r_(1)x)+C_(2)e^(r_(2)x)+dots+C_(n)e^(r_(n)x)+(Ae^([alpha-(r_(1)+r_(2)+dots+r_(n))]x))/(C_(1)C_(2)dotsC_(n)V^(2)[r_(1),r_(2),dots,r_(n),alpha-(r_(1)+r_(2)+dots+r_(n))])y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}+\ldots+C_{n} e^{r_{n} x}+\frac{A e^{\left[\alpha-\left(r_{1}+r_{2}+\ldots+r_{n}\right)\right] x}}{C_{1} C_{2} \ldots C_{n} V^{2}\left[r_{1}, r_{2}, \ldots, r_{n}, \alpha-\left(r_{1}+r_{2}+\ldots+r_{n}\right)\right]}
y=C_(1)e^(r_(1)x)+C_(2)e^(r_(2)x)+dots+C_(n)e^(r_(n)x)+(Ae^([alpha-(r_(1)+r_(2)+dots+r_(n))]x))/(C_(1)C_(2)dotsC_(n)V^(2)[r_(1),r_(2),dots,r_(n),alpha-(r_(1)+r_(2)+dots+r_(n))])y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}+\ldots+C_{n} e^{r_{n} x}+\frac{A e^{\left[\alpha-\left(r_{1}+r_{2}+\ldots+r_{n}\right)\right] x}}{C_{1} C_{2} \ldots C_{n} V^{2}\left[r_{1}, r_{2}, \ldots, r_{n}, \alpha-\left(r_{1}+r_{2}+\ldots+r_{n}\right)\right]}
However, let us also consider the integrals of the differential equation (31) of the form (17), that is
{:[y=[(C_(1))/((p_(1)-1)!)x^(p_(1)-1)+(C_(1,1))/((p_(1)-2)!)x^(p_(1)-2)+dots+C_(1,p_(1)-1)]e^(r_(2)x)],[+cdots cdots+cdots***********],[+[(C_(k-1))/((p_(k-1)-1)!)x^(p_(k-1)-1)+(C_(k-1,1))/((p_(k-1)-2)!)x^(p_(k-1)-2)+dots+C_(k-1,p_(k-1)-1)]e^(r_(k-1)x)],[(33)+[(C_(k))/((p_(k)-1)!)x^(p_(k)-1)+(C_(k-1,1))/((p_(k)-2)!)x^(p_(k)-2)+dots+C_(k,p_(k)-1)]e^(r_(k)x)]:}\begin{align*}
y & =\left[\frac{C_{1}}{\left(p_{1}-1\right)!} x^{p_{1}-1}+\frac{C_{1,1}}{\left(p_{1}-2\right)!} x^{p_{1}-2}+\ldots+C_{1, p_{1}-1}\right] e^{r_{2} x} \\
& +\cdots \cdots+\cdots \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \\
& +\left[\frac{C_{k-1}}{\left(p_{k-1}-1\right)!} x^{p_{k-1}-1}+\frac{C_{k-1,1}}{\left(p_{k-1}-2\right)!} x^{p_{k-1}-2}+\ldots+C_{k-1, p_{k-1}-1}\right] e^{r_{k-1} x} \\
& +\left[\frac{C_{k}}{\left(p_{k}-1\right)!} x^{p_{k}-1}+\frac{C_{k-1,1}}{\left(p_{k}-2\right)!} x^{p_{k}-2}+\ldots+C_{k, p_{k}-1}\right] e^{r_{k} x} \tag{33}
\end{align*}
where
{:(34)p_(1)+p_(2)+dots+p_(k)=n+1:}\begin{equation*}
p_{1}+p_{2}+\ldots+p_{k}=n+1 \tag{34}
\end{equation*}
andr_(1),r_(2),dots,r_(k)r_{1}, r_{2}, \ldots, r_{k}are distinct. According to formula (18), we have
Delta_(n)[y]=(-1)^(k)C_(1)^(p_(1))C_(2)^(p_(2))dotsC_(k-1)^(p_(k-1))C_(k)^(p_(k))V^(2)(ubrace(r_(1),dots,r_(1))_(p_(1)" ori "),dots,ubrace(r_(k),dots,r_(k))_(p_(1)" ori "))e^((p_(1)r_(1)+dots+p_(k)r_(Lambda))x)\Delta_{n}[y]=(-1)^{k} C_{1}^{p_{1}} C_{2}^{p_{2}} \ldots C_{k-1}^{p_{k-1}} C_{k}^{p_{k}} V^{2}(\underbrace{r_{1}, \ldots, r_{1}}_{p_{1} \text { ori }}, \ldots, \underbrace{r_{k}, \ldots, r_{k}}_{p_{1} \text { ori }}) e^{\left(p_{1} r_{1}+\ldots+p_{k} r_{\Lambda}\right) x}
For the second member to reduce toAe^(alpha x)A e^{\alpha x}, we will choose
{:[p_(1)r_(1)+dots+p_(k-1)r_(k-1)+p_(k)r_(k)=alpha],[(-1)^(k)C_(1)^(p_(1))C_(2)^(p_(2))dotsC_(k)^(p_(k))V^(2)(ubrace(r_(1),dots,r_(1))_(p_(1)" ori ")","dots","ubrace(r_(k),dots,r_(k))_(p_(k)" ori "))=A]:}\begin{gathered}
p_{1} r_{1}+\ldots+p_{k-1} r_{k-1}+p_{k} r_{k}=\alpha \\
(-1)^{k} C_{1}^{p_{1}} C_{2}^{p_{2}} \ldots C_{k}^{p_{k}} V^{2}(\underbrace{r_{1}, \ldots, r_{1}}_{p_{1} \text { ori }}, \ldots, \underbrace{r_{k}, \ldots, r_{k}}_{p_{k} \text { ori }})=A
\end{gathered}
where the numbersp_(1),p_(2),dots,p_(k-1),p_(k)p_{1}, p_{2}, \ldots, p_{k-1}, p_{k}are fixed so that we have the relation (34) and thatp_(k)!=0p_{k} \neq 0.
Then we will take
{:(35)r_(k)=(alpha-(p_(1)r_(1)+dots+p_(k-1)r_(k-1)))/(p_(k)):}\begin{equation*}
r_{k}=\frac{\alpha-\left(p_{1} r_{1}+\ldots+p_{k-1} r_{k-1}\right)}{p_{k}} \tag{35}
\end{equation*}
the numbersr_(1),r_(2),dots,r_(k-1)r_{1}, r_{2}, \ldots, r_{k-1}being chosen so that all numbersr_(1),r_(2),dots,r_(k)r_{1}, r_{2}, \ldots, r_{k}to be distinct.
Next we will take
C_(k)=root(p_(k))((A)/((-1)^(k)C_(1)^(p_(1))C_(2)^(p_(2))dotsC_(k-1)^(p_(k-1))V^(2)(ubrace(r_(1),dots,r_(1))_(p_(1)" ori "),dots,r_(p_(k),dots,r_(k))^(r_(k)))))C_{k}=\sqrt[p_{k}]{\frac{A}{(-1)^{k} C_{1}^{p_{1}} C_{2}^{p_{2}} \ldots C_{k-1}^{p_{k-1}} V^{2}(\underbrace{r_{1}, \ldots, r_{1}}_{p_{1} \text { ori }}, \ldots, r_{p_{k}, \ldots, r_{k}}^{r_{k}})}}
and then the differential equation (29) still has the integrals given by formula (33), wherer_(k)r_{k}andC_(k)C_{k}are given by formulas (35) and (36).
Examples.
1^(@)1^{\circ}. The equation of the chain. This is
Delta[y]=|[y,y^(')],[y^('),y^('')]|=1\Delta[y]=\left|\begin{array}{ll}
y & y^{\prime} \\
y^{\prime} & y^{\prime \prime}
\end{array}\right|=1
Its integral is found among the integrals of the differential equation
Delta_(2)[y]=0\Delta_{2}[y]=0
This equation has integrals of the form
{:(37)y=C_(1)e^(r_(1)x)+C_(2)e^(r_(2)x):}\begin{equation*}
y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x} \tag{37}
\end{equation*}
and
{:(38)y=(C_(1)x+C_(2))e^(r_(1)x):}\begin{equation*}
y=\left(C_{1} x+C_{2}\right) e^{r_{1} x} \tag{38}
\end{equation*}
For these integrals we have
{:[Delta_(1)[y]=C_(1)C_(2)(r_(1)-r_(2))^(2)e^((r_(1)+r_(2))x)],[Delta_(1)[y]=-C_(1)^(2)e^(2r_(1)x).]:}\begin{aligned}
& \Delta_{1}[y]=C_{1} C_{2}\left(r_{1}-r_{2}\right)^{2} e^{\left(r_{1}+r_{2}\right) x} \\
& \Delta_{1}[y]=-C_{1}^{2} e^{2 r_{1} x} .
\end{aligned}
For integrals of the form (37), so that we haveDelta_(1)[y]=1\Delta_{1}[y]=1we will choose
r_(1)+r_(2)=0,quadr_(1)-r_(2)=(1)/(sqrt(C_(1)C_(2)))r_{1}+r_{2}=0, \quad r_{1}-r_{2}=\frac{1}{\sqrt{C_{1} C_{2}}}
which leads us to
{:('")"y=C_(1)e^((x)/(2sqrt(C_(1)C_(2))))+C_(2)e^(-(x)/(2sqrt(C_(1)C_(2)))):}\begin{equation*}
y=C_{1} e^{\frac{x}{2 \sqrt{C_{1} C_{2}}}}+C_{2} e^{-\frac{x}{2 \sqrt{C_{1} C_{2}}}} \tag{$\prime$}
\end{equation*}
For integrals of the form (38), so that we haveDelta_(1)[y]=1\Delta_{1}[y]=1we will chooser_(1)=0,C_(1)=ir_{1}=0, \mathrm{C}_{1}=i, which leads us to
{:(38')y=ix+C_(2):}\begin{equation*}
y=i x+C_{2} \tag{38'}
\end{equation*}
The integrals of the differential equation of the chain are(37^('))\left(37^{\prime}\right)and(38^('))\left(38^{\prime}\right).
2^(@)2^{\circ}. Darboux equation. It is the equation
2^(@)2^{\circ}. Darboux equation. It is the equation
Delta_(2)[y]=|[y,y^('),y^('')],[y^('),y^(''),y^(''')],[y^(''),y^('''),y^('v)]|=1\Delta_{2}[y]=\left|\begin{array}{ccc}
y & y^{\prime} & y^{\prime \prime} \\
y^{\prime} & y^{\prime \prime} & y^{\prime \prime \prime} \\
y^{\prime \prime} & y^{\prime \prime \prime} & y^{\prime v}
\end{array}\right|=1
Its integrals are found among the integrals of the equation
which are of the form
which are of the form
Delta_(3)[y]=0\Delta_{3}[y]=0
{:[(39)y=C_(1)e^(r_(1)x)+C_(2)e^(r_(2)x)+C_(3)e^(r_(3)x)],[(40)y=(C_(1)x+C_(2))e^(r_(1)x)+C_(3)e^(r_(3)x)],[(41)y=(C_(1)(x^(2))/(2)+C_(2)x+C_(3))e^(r_(1)x)]:}\begin{align*}
& y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}+C_{3} e^{r_{3} x} \tag{39}\\
& y=\left(C_{1} x+C_{2}\right) e^{r_{1} x}+C_{3} e^{r_{3} x} \tag{40}\\
& y=\left(C_{1} \frac{x^{2}}{2}+C_{2} x+C_{3}\right) e^{r_{1} x} \tag{41}
\end{align*}
For these types of integrals, we have
{:[Delta_(2)[y]=C_(1)C_(2)C_(3)V^(2)(r_(1),r_(2),r_(3))e^((r_(1)+r_(2)+r_(3))x)],[Delta_(2)[y]=-C_(1)^(2)C_(3)V^(2)(r_(1),r_(1),r_(3))e^((2r_(1)+r_(3))x)],[Delta_(2)[y]=-C_(1)^(3)e^(3r_(1)x)]:}\begin{aligned}
& \Delta_{2}[y]=C_{1} C_{2} C_{3} V^{2}\left(r_{1}, r_{2}, r_{3}\right) e^{\left(r_{1}+r_{2}+r_{3}\right) x} \\
& \Delta_{2}[y]=-C_{1}^{2} C_{3} V^{2}\left(r_{1}, r_{1}, r_{3}\right) e^{\left(2 r_{1}+r_{3}\right) x} \\
& \Delta_{2}[y]=-C_{1}^{3} e^{3 r_{1} x}
\end{aligned}
BecauseDelta_(2)[y]\Delta_{2}[y]to be equal to 1, we will choose in the first case
r_(3)=-(r_(1)+r_(2)),C_(3)=(1)/(C_(1)C_(2)V^(2)[r_(1),r_(2),-(r_(1)+r_(2))])r_{3}=-\left(r_{1}+r_{2}\right), C_{3}=\frac{1}{C_{1} C_{2} V^{2}\left[r_{1}, r_{2},-\left(r_{1}+r_{2}\right)\right]}
in the second case
r_(3)=-2r_(1),quadC_(3)=-(1)/(C_(1)^(2)V^(2)(r_(1),r_(1),-2r_(1)))r_{3}=-2 r_{1}, \quad C_{3}=-\frac{1}{C_{1}^{2} V^{2}\left(r_{1}, r_{1},-2 r_{1}\right)}
and in the third case
r_(1)=0quadC_(1)=-1r_{1}=0 \quad C_{1}=-1
This leads us to the following integrals of Darboux's equation:
{:[(39)y=C_(1)e^(r_(1)x)+C_(2)e^(r_(2)x)+(e^(-(r_(1)+r_(2))x))/(C_(1)C_(2)(r_(2)-r_(1))^(2)(2r_(1)+r_(2))^(2)(r_(1)+2r_(2))^(2))],[('")"y=(C_(1)x+C_(2))e^(r_(1)x)-(e^(-2r_(1)x))/(81C_(1)^(2)r_(1)^(4))],[(41́)y=-(x^(2))/(2)+C_(2)x+C_(3)]:}\begin{align*}
& y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}+\frac{e^{-\left(r_{1}+r_{2}\right) x}}{C_{1} C_{2}\left(r_{2}-r_{1}\right)^{2}\left(2 r_{1}+r_{2}\right)^{2}\left(r_{1}+2 r_{2}\right)^{2}} \tag{39}\\
& y=\left(C_{1} x+C_{2}\right) e^{r_{1} x}-\frac{e^{-2 r_{1} x}}{81 C_{1}^{2} r_{1}^{4}} \tag{$\prime$}\\
& y=-\frac{x^{2}}{2}+C_{2} x+C_{3} \tag{41́}
\end{align*}
BIBLIOGRAPHY
- H. Löwner, Über monotone Matrixfunctionen. Math. Zeit., 38 (1934) 177-216.
- G. Darboux, Sur une équation différenceielle du quatriòme ordre. CR de l'Ac. often Sci. de Paris, vol. CXLI, pp. 415-417.
-
- Sur une équation differential du quatrième ordre, CR de l'Ac. often Sci. de Paris, vol. CXLI, pp. 483-484.
- AG Kuros, Course in Higher Algebra. Technical Publishing House, Buc., 1955, p. 133.
- Th. J. Stieltjes, Quelques recherches sur la théorie des quadratures dites mécaniques. Ann. of the Ecole Normale Supérieure, 1884, pp. 409-426.
- DV Ionescu, Numerical Quadratures. Technical Publishing House, Buc., 1957, p. 262.
Integration of one differential equation
(Brief content)
In this work, the differential equation is integrated, (1)Delta_(n)[y]=0\Delta_{n}[y]=0and differential equation (2)Delta_(n)[y]=Ae^(alpha x)\Delta_{n}[y]=A e^{\alpha x}, where A andalpha\alphaconstants. EquationDelta_(1)[y]=1\Delta_{1}[y]=1it is the differential equation of the chain line, and the equationDelta_(2)[y]=1\Delta_{2}[y]=1was studied by G. Darbu[1,2][1,2].
In this work, the differential equation is integrated, (1)Delta_(n)[y]=0\Delta_{n}[y]=0and differential equation (2)Delta_(n)[y]=Ae^(alpha x)\Delta_{n}[y]=A e^{\alpha x}, where A andalpha\alphaconstants. EquationDelta_(1)[y]=1\Delta_{1}[y]=1it is the differential equation of the chain line, and the equationDelta_(2)[y]=1\Delta_{2}[y]=1was studied by G. Darbu[1,2][1,2].
Integral equationsDelta_(n)^(F)[y]=A\Delta_{n}^{F}[y]=Afor whichDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0, given by formulas (10) and (17) and they are true in the entire interval (-oo,+oo-\infty,+\infty).
For these integrals, we have formulas (12) and (18), which are true in the interval(-oo,+oo)(-\infty,+\infty).
It is proved that any integral equationDelta_(q)[y]=0\Delta_{q}[y]=0withDelta_(q-1)[y]!=0\Delta_{q-1}[y] \neq 0whereq < nq<n, is also integralDelta_(n)[y]=0\Delta_{n}[y]=0.
With the help of these results, all the integrals of the differential equation are obtainedDelta_(n)[y]=0\Delta_{n}[y]=0.
As an application, the differential equation (2) is integrated, preliminarily proving the identity (30), which reduces the finding of the integrals of equation (2) to the integration of the equationDelta_(n+1)[y]=0\Delta_{n+1}[y]=0. The results are given by formulas (32) and (33), whereΥ_(k)\Upsilon_{k}andC_(k)C_{k}given by formulas (35) and (36).
The integrals of Darbou's differential equation (4) are given by formulas (39'), (40'), (41').
The integration of a differential equation
(Résumé)
Dans ce mémoire on integré l'équation différenceielle(1),Delta_(n)[y]=0\Delta_{n}[y]=0and the differential equation (2)Delta_(n)[y]=Aepsilon^(alpha r)\Delta_{n}[y]=A \epsilon^{\alpha r}, whereAAandalpha\alphasont des constantes L'équationDelta_(1)[y]=1\Delta_{1}[y]=1est l'équation différenceielle de la chaînette et l'équationDelta_(2)[y]=1\Delta_{2}[y]=1was studied by G. Darboux[1,2][1,2].
Dans ce mémoire on integré l'équation différenceielle(1),Delta_(n)[y]=0\Delta_{n}[y]=0and the differential equation (2)Delta_(n)[y]=Aepsilon^(alpha r)\Delta_{n}[y]=A \epsilon^{\alpha r}, whereAAandalpha\alphasont des constantes L'équationDelta_(1)[y]=1\Delta_{1}[y]=1est l'équation différenceielle de la chaînette et l'équationDelta_(2)[y]=1\Delta_{2}[y]=1was studied by G. Darboux[1,2][1,2].
Les intégrales de l'équationDelta_(n)[y]=0\Delta_{n}[y]=0for whichDelta_(n-1)[y]!=0\Delta_{n-1}[y] \neq 0sont données par les formulas (10) et (17) valid dans l'intervale (-oo-\infty,+oo)+\infty). Pour ces intégrales on a les formulas (12) et (18), valid dans 1'intervale (-oo,+oo-\infty,+\infty).
On demune aussi que toute intégrale de l'équationDelta_(q)[y]=0\Delta_{q}[y]=0, withDelta_(q-1)[y]!=0\Delta_{q-1}[y] \neq 0, whereq < nq<n, est également intégrale de l'équationDelta_(n)[y]=0\Delta_{n}[y]=0.
A l'aide de ces résultats we obtain toutes les integrales de l'équation différenceielleDelta_(n)[y]=0\Delta_{n}[y]=0.
Comme application on integré l'équation différenceielle (2), en démontrant au préalé l'identité (30) qui amène la recherche des integrales de l'équation (2) à l'intégration de l'équationDelta_(n+1)[y]=0\Delta_{n+1}[y]=0. The results are given by formulas (32) and (33) whereΥ_(k)\Upsilon_{k}andC_(k)C_{k}are given by formulas (35) and (36).
The integrals of the differential equation (4) of Darboux are given by the formulas(39^(')),(40^(')),(41^('))\left(39^{\prime}\right),\left(40^{\prime}\right),\left(41^{\prime}\right).