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UDC 517.934.92
INTEGRABILITY CONDITIONS IN FINITE FORM FOR CERTAIN SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER
V. G. Skatetskii
Consider the system
\[ \frac{dX}{dt}=X\left(U_0\varphi_0+U_1\varphi_1+U_0U_1\varphi_2+U_1U_0\varphi_3\right), \tag{1} \]
where \(U_i\) \((i=0,1)\) are matrices of the second order, whose determinants and traces \(\sigma(U_i)\) are equal to zero; \(\varphi_i\) \((i=0,1,2,3)\) are continuous scalar functions of \(t\).
We shall show that if the functions \(\varphi_0\) and \(\varphi_1\) satisfy the condition
\[ \varphi_0=-\frac{\sigma_{01}}{\sigma_0}(\varphi_2+\varphi_3), \qquad \varphi_1=-\frac{\sigma_0}{(a+b)^2}(a^2\varphi_2+b^2\varphi_3), \tag{2} \]
where \(\sigma_0,\sigma_{01}\) are the nonzero traces of the corresponding matrices \(U_0, U_0U_1\), and \(a\) and \(b\) are arbitrary numbers satisfying the condition \(a+b\ne0\), \(a\ne0\), \(b\ne0\), then system (1) has a solution of the form
\[ X=\exp\left[A\int_{t_0}^{t} F_1\exp(2L_2)\,d\tau\right]\exp(BL_2), \tag{3} \]
where
\[ L_2=\int_{t_0}^{t}F_2\,d\tau; \]
\[ A=-\frac{(a+b)\sigma_{01}}{\sigma_0}U_0-\frac{ab\sigma_0}{a+b}U_1+bU_0U_1+aU_1U_0, \]
\[ B=\frac{a+b}{b\sigma_0}U_0+\frac{b\sigma_0}{(a+b)\sigma_{01}}U_1-\frac{a+b}{b\sigma_{01}}U_1U_0, \tag{4} \]
\[ F_1=\frac{\varphi_2}{b}, \qquad F_2=\frac{\sigma_{01}}{a+b}(a\varphi_2-b\varphi_3). \tag{5} \]
We shall carry out the proof in the same way as in [1]. We shall show that the identity
\[ 2AB\dot{A}=A^2B+\dot{B}A^2 \tag{6} \]
holds.
Indeed, by computation, using the properties of the matrices \(U_0\) and \(U_1\), considered in [2], one can show that \(A^2=0\), \(BA=A\), \(AB=-A\). Hence it is clear that identity (6) holds.
On the basis of [3] (pp. 52–53), the system
\[ \frac{dX}{dt}=X(AF_1+BF_2), \tag{7} \]
where \(A\) and \(B\) are determined by formulas (4); in this case there will be a solution of the form (3).
System (7) can be rewritten as
\[
\frac{dX}{dt}
=
X\left[
\frac{a+b}{\sigma_0}
\left(
-\sigma_{01}F_1+\frac{1}{b}F_2
\right)U_0
+
\frac{b\sigma_0}{a+b}
\left(
-aF_1+\frac{1}{\sigma_{01}}F_2
\right)U_1
+
\right.
\]
\[
\left.
+
bF_1U_0U_1
+
\left(
aF_1-\frac{a+b}{b\sigma_{01}}F_2
\right)U_1U_0
\right].
\tag{8}
\]
In order that system (1) be equivalent to system (8), it is necessary and sufficient that the following equalities hold:
\[ \frac{a+b}{\sigma_0} \left( -\sigma_{01}F_1+\frac{1}{b}F_2 \right)=\varphi_0, \qquad \frac{b\sigma_0}{a+b} \left( -aF_1+\frac{1}{\sigma_{01}}F_2 \right)=\varphi_1, \]
\[ bF_1=\varphi_2, \qquad aF_1-\frac{a+b}{b\sigma_{01}}F_2=\varphi_3. \]
From the last two equalities we obtain expression (5). Substituting the obtained values of \(F_1\) and \(F_2\) into the first two equalities of the latter relations, we obtain the sufficient conditions (2).
Let us now consider a system of the following form:
\[ \frac{dX}{dt}=X(U_0\varphi_0+U_1\varphi_1+U_2\varphi_2), \tag{9} \]
where \(U_i\) \((i=0,1,2)\) are matrices of the second order whose determinants and the trace \(\sigma(U_1)\) are equal to zero; \(\varphi_i\) \((i=0,1,2)\) are continuous scalar functions of \(t\); \(\sigma_0\ne0\), \(\sigma_{01}\ne0\). According to [2] and taking into account the identities
\[ U_1U_0=-U_0U_1+\sigma_0U_1+\sigma_{01}, \]
\[ U_2= \frac{\sigma_{12}}{\sigma_{01}}U_0 + \frac{\sigma_{01}\sigma_{02}-\sigma_0(\sigma_{012}+\sigma_{102})}{\sigma_{01}^{2}}U_1 + \]
\[ + \frac{\sigma_{012}U_0U_1+\sigma_{102}U_1U_0}{\sigma_{01}^{2}} + \sigma_2-\frac{\sigma_{012}+\sigma_{102}}{\sigma_{01}}, \]
considered in [2], system (9) can be written as
\[ \frac{dX}{dt} = X\left\{ \left(\varphi_0+\frac{\sigma_{12}}{\sigma_{01}}\varphi_2\right)U_0 + \left(\varphi_1+ \frac{\sigma_{01}\sigma_{02}-\sigma_0\sigma_{012}}{\sigma_{01}^{2}}\varphi_2 \right)U_1 + \right. \]
\[ \left. + \frac{\sigma_{012}-\sigma_{102}}{\sigma_{01}^{2}}\varphi_2U_0U_1 + \left(\sigma_2-\frac{\sigma_{012}}{\sigma_{01}}\right)\varphi_2 \right\}, \]
where \(\sigma_{12}\), \(\sigma_{02}\), \(\sigma_{012}\), and \(\sigma_{102}\) are the traces of the corresponding matrices \(U_1U_2\), \(U_0U_2\), \(U_0U_1U_2\), and \(U_1U_0U_2\). Applying to the last system the substitution
\[ X = Y\exp\left[ \left(\sigma_2-\frac{\sigma_{012}}{\sigma_{01}}\right) \int_{t_0}^{t}\varphi_2\,dt \right] \]
under the condition that
\[ \sigma_2\ne\frac{\sigma_{012}}{\sigma_{01}}, \]
we obtain
\[ \frac{dY}{dt}=Y\left\{\left(\varphi_0+\frac{\sigma_{12}}{\sigma_{01}}\varphi_2\right)U_0+ \left(\varphi_1+\frac{\sigma_{01}\sigma_{02}-\sigma_0\sigma_{012}}{\sigma_{01}^{2}}\varphi_2\right)U_1+ \frac{\sigma_{012}-\sigma_{102}}{\sigma_{01}^{2}}\varphi_2U_0U_1\right\}. \tag{10} \]
Arguing in the same way, taking into account for the matrices \(U_0\) and \(U_1\) the identities indicated above and applying the substitution
\[ X=Y\exp\left[\int_{t_0}^{t}\left(a\sigma_{01}F_1-\frac{a+b}{b}F_2\right)d\tau\right], \]
in the present case the system (7) can be transformed to the form
\[ \frac{dY}{dt}=Y\left\{\left[-\frac{(a+b)\sigma_{01}}{\sigma_0}F_1+ \frac{a+b}{b\sigma_0}F_2\right]U_0+\right. \]
\[ \left.+\left[\frac{a^2\sigma_0}{a+b}F_1- \frac{a\sigma_0(a+2b)}{b(a+b)\sigma_{01}}F_2\right]U_1+ \left[(b-a)F_1+\frac{a+b}{b\sigma_{01}}F_2\right]U_0U_1\right\}. \tag{11} \]
The solution of this system has the form
\[ Y=X\exp\left[-\int_{t_0}^{t}\left(a\sigma_{01}F_1-\frac{a+b}{b}F_2\right)d\tau\right], \tag{12} \]
where \(X\) is determined by formula (3).
In order that system (10) be equivalent to system (11), it is necessary and sufficient that the equalities
\[ \frac{a+b}{\sigma_0}\left(-\sigma_{01}F_1+\frac{1}{b}F_2\right) =\varphi_0+\frac{\sigma_{12}}{\sigma_{01}}\varphi_2, \]
\[ \frac{a\sigma_0}{a+b}\left(aF_1-\frac{a+2b}{b\sigma_{01}}F_2\right) =\varphi_1+\frac{\sigma_{01}\sigma_{02}-\sigma_0\sigma_{012}}{\sigma_{01}^{2}}\varphi_2, \tag{13} \]
\[ (b-a)F_1+\frac{a+b}{b\sigma_{01}}F_2 =\frac{\sigma_{012}-\sigma_{102}}{\sigma_{01}^{2}}\varphi_2. \]
Finding \(F_1\) and \(F_2\) from the first two equations of system (13),
\[ F_1=-\frac{(a+2b)\sigma_0}{2b(a+b)\sigma_{01}}\varphi_0 -\frac{a+b}{2ab\sigma_0}\varphi_1+ \]
\[ +\frac{(a+b)^2(\sigma_0\sigma_{012}-\sigma_{01}\sigma_{02})-\sigma_0^2\sigma_{12}a(a+2b)} {2ab(a+b)\sigma_0\sigma_{01}^{2}}\varphi_2, \tag{14} \]
\[ F_2=-\frac{a\sigma_0}{2(a+b)}\varphi_0 -\frac{\sigma_{01}(a+b)}{2a\sigma_0}\varphi_1- \]
\[ -\frac{(a+b)^2(\sigma_{01}\sigma_{02}-\sigma_0\sigma_{012})+a^2\sigma_0^2\sigma_{12}} {2a(a+b)\sigma_0\sigma_{01}}\varphi_2 \]
and substituting them into the third equation of system (13), we obtain that
\[ \varphi_0=-\frac{(a+b)^2\sigma_{01}}{ab\sigma_0^2}\,\varphi_1- \]
\[ -\frac{ab\sigma_0^2\sigma_{12}+(a+b)\left[(a+b)\sigma_{01}\sigma_{02}-\sigma_0(a\sigma_{102}+b\sigma_{012})\right]}{ab\sigma_0^2\sigma_{01}}\,\varphi_2 . \tag{15} \]
Thus, the following result has been obtained: if \(\varphi_0\) satisfies condition (15), then system (10) has a solution of the form (12), where \(F_1\) and \(F_2\) are determined by formulas (14), and \(A\) and \(B\) by formulas (4).
Remark. Let a system be given
\[ \frac{dX}{dt}=X\sum_{i=0}^{n}U_i\varphi_i, \tag{16} \]
where \(U_i\) \((i=0,1,\ldots,n)\) are matrices of the second order, the determinants of which and the trace of one of them are equal to zero. Without loss of generality, we assume that \(\sigma(U_1)=0\), and that \(\varphi_i\) are continuous scalar functions of \(t\). As before, we assume that \(\sigma_0\ne0\) and \(\sigma_{01}\ne0\). According to [1] and [2], this system can be written in the following form:
\[ \frac{dY}{dt} = Y\left\{ \left(\varphi_0+\sum_{i=2}^{n}\frac{\sigma_{1i}}{\sigma_{01}}\varphi_i\right)U_0+ \right. \]
\[ \left. +\left(\varphi_1+\sum_{i=2}^{n}\frac{\sigma_{01}\sigma_{0i}-\sigma_0\sigma_{01i}}{\sigma_{01}^{2}}\varphi_i\right)U_1 +\sum_{i=2}^{n}\frac{\sigma_{01i}-\sigma_{10i}}{\sigma_{01}^{2}}\varphi_i U_0U_1 \right\}, \]
where \(\sigma_{1i}\), \(\sigma_{01i}\), and \(\sigma_{10i}\) are the traces of the corresponding matrices \(U_1U_i\), \(U_0U_1U_i\), and \(U_1U_0U_i\). System (16) is integrable in finite form if the relation
\[ \varphi_0=-\frac{(a+b)^2\sigma_{01}}{ab\sigma_0^2}\,\varphi_1- \]
\[ -\sum_{i=2}^{n} \frac{ab\sigma_0^2\sigma_{0i}+(a+b)\left[(a+b)\sigma_{01}\sigma_{0i}-\sigma_0(a\sigma_{10i}+b\sigma_{01i})\right]} {ab\sigma_0^2\sigma_{01}}\, \varphi_i, \]
is satisfied, and the integral matrix \(Y\) is determined by formula (12), where \(F_1\) and \(F_2\) have the form:
\[ F_1 = -\frac{(a+2b)\sigma_0}{2b(a+b)\sigma_{01}}\varphi_0 -\frac{a+b}{2ab\sigma_0}\varphi_1+ \]
\[ +\sum_{i=2}^{n} \frac{(a+b)^2(\sigma_0\sigma_{01i}-\sigma_{01}\sigma_{0i})-\sigma_0^2\sigma_{1i}a(a+2b)} {2ab(a+b)\sigma_0\sigma_{01}^{2}}\, \varphi_i, \]
\[ F_2 = -\frac{a\sigma_0}{2(a+b)}\varphi_0 -\frac{(a+b)\sigma_{01}}{2a\sigma_0}\varphi_1- \]
\[ -\sum_{i=2}^{n} \frac{(a+b)^2(\sigma_{01}\sigma_{0i}-\sigma_0\sigma_{01i})+a^2\sigma_0^2\sigma_{1i}} {2a(a+b)\sigma_0\sigma_{01}}\, \varphi_i . \]
References
- Fedorov G. F. Vestnik LGU, No. 11, 57–65, 1953.
- Fedorov G. F. Izv. vuzov, No. 3, 217–224, 1958.
- Erugin N. P. Linear Systems of Ordinary Differential Equations. Minsk, 1963.
Received by the editors
November 19, 1965
Institute of Mathematics of the Academy of Sciences of the BSSR