Brief Communications

UDC 517.941.92

Solution in Finite Form of Certain Systems of Ordinary Differential Equations of Second Order

V. G. Skatetskii

A system is given

\[ \frac{dX}{dt}=X\left(U_0\varphi_0+U_1\varphi_1+U_0U_1\varphi_2+U_1U_0\varphi_3\right), \tag{1} \]

where \(U_i\) \((i=0,1)\) are constant matrices of second order with \(\det U_i=0\) \((i=0,1)\); \(\varphi_i\) \((i=0,1,2,3)\) are continuous scalar functions of \(t\).

We shall find a sufficient condition under which the solution of system (1) can be obtained in finite form. To this end, consider the system

\[ \frac{dX}{dt}=X\left(AF_1+BF_2\right), \tag{2} \]

where \(A\) and \(B\) are constant matrices of second order; \(F_1\) and \(F_2\) are continuous scalar functions of \(t\). In [1] (pp. 52, 53) it is shown that if the matrix \(A\) has relatively prime elementary divisors and the condition

\[ [A[BA]]=0, \tag{3} \]

is satisfied, then system (2) has a solution in finite form

\[ X=\exp\left[\int_{t_0}^{t}\exp(BL_2)A\exp(-BL_2)F_1\,d\tau\right]\exp(BL_2), \tag{4} \]

where

\[ L_2=\int_{t_0}^{t}F_2\,d\tau. \]

Let the matrix \(A\) have the form

\[ A=a_1U_0+\frac{a_2^{\,2}\sigma_{01}}{a_1}\,U_1+a_2\left(U_0U_1+U_1U_0\right), \tag{5} \]

where \(a_1\) and \(a_2\) are arbitrary nonzero numbers, and \(\sigma_{01}\) is the trace of the matrices \(U_0U_1\) and \(U_1U_0\). We assume that \(\sigma_{01}\ne0\). By calculation and using the properties of the matrices \(U_0\) and \(U_1\), considered in [2] § 1, one can show that

\[ A^2=\sigma_A A, \tag{6} \]

where \(\sigma_A\) is the nonzero trace of the matrix \(A\).

Construct the matrix \(B\) so that identity (3) is satisfied, or, applying condition (6), identity (326) of I. A. Lappo-Danilevskii [3] (p. 102),

\[ 2\sigma_{AB}A=\sigma_A(AB+BA), \tag{7} \]

where \(\sigma_{AB}\) is the trace of the matrix \(AB\). In order that identity (7) be satisfied, it is sufficient that

\[ AB+BA=0\quad \text{and}\quad \sigma_{AB}=0. \tag{8} \]

We shall seek \(B\) in the form
\[ B=\alpha U_0+\beta U_1+\gamma U_0U_1+\delta U_1U_0, \]
where \(\alpha,\beta,\gamma\), and \(\delta\) are the quantities to be found. Using the properties of the matrices \(U_0\) and \(U_1\), we obtain that

\[ \begin{aligned} AB+BA={}&[2M\alpha+\sigma_{01}N(\gamma+\delta)]U_0+\\ &+\frac{a_2\sigma_{01}}{a_1}[2N\beta+M(\gamma+\delta)]U_1 +\left(\frac{a_2}{a_1}M\alpha+N\beta+\sigma_A\gamma\right)U_0U_1+\\ &+\left(\frac{a_2}{a_1}M\alpha+N\beta+\sigma_A\delta\right)U_1U_0, \end{aligned} \]

and

\[ \sigma_{AB}=\frac{1}{a_1}\,[M^2\alpha+\sigma_{01}N^2\beta+\sigma_{01}MN(\gamma+\delta)], \]

where \(M=a_1\sigma_0+a_2\sigma_{01}\ne0\) and \(N=a_1+a_2\sigma_1\ne0\). In these notations
\[ \sigma_A=M+\frac{a_2\sigma_{01}}{a_1}N. \]

In order that \(AB+BA=0\), it is sufficient that the coefficients of \(U_0,\ U_1,\ U_0U_1\), and \(U_1U_0\) be equal to zero. A necessary condition for the last two equations of the system obtained to have a place is \(\gamma=\delta\). Hence we have

\[ M\alpha+\sigma_{01}N\gamma=0,\qquad N\beta+M\gamma=0,\qquad a_2M\alpha+a_1N\beta+a_1\sigma_A\gamma=0. \]

It is not difficult to see that the last equation is a consequence of the first two. Therefore
\[ \alpha=-\frac{\sigma_{01}N}{M}\gamma,\qquad \beta=-\frac{M}{N}\gamma. \]
For these values of \(\alpha,\beta\), and \(\gamma=\delta\), we have \(\sigma_{AB}=0\).

Thus, the sufficient condition (7) is satisfied. In what follows we shall assume that \(\gamma=1\). Then

\[ B=-\frac{\sigma_{01}N}{M}U_0-\frac{M}{N}U_1+U_0U_1+U_1U_0. \tag{9} \]

Substituting the values of \(A\) and \(B\) into (2), we obtain

\[ \frac{dX}{dt} = X\left[ U_0\left(a_1F_1-\frac{\sigma_{01}N}{M}F_2\right) + U_1\left(\frac{a_2^2\sigma_{01}}{a_1}F_1-\frac{M}{N}F_2\right) + (U_0U_1+U_1U_0)(a_2F_1+F_2) \right]. \tag{10} \]

In order that system (1) be equivalent to system (10), it is necessary and sufficient that the following equalities hold:

\[ a_1F_1-\frac{\sigma_{01}N}{M}F_2=\varphi_0,\qquad \frac{a_2^2\sigma_{01}}{a_1}F_1-\frac{M}{N}F_2=\varphi_1, \]

\[ a_2F_1+F_2=\varphi_2,\qquad \varphi_2=\varphi_3. \tag{11} \]

Assume that the determinant
\[ \Delta_F=\frac{d\sigma_A}{MN}, \]
where
\[ d=a_2^2\sigma_{01}\sigma_1-a_1^2\sigma_0, \]
formed from the coefficients of \(F_1\) and \(F_2\) in the first two equations, is not equal to zero. Then, solving the first two equations jointly, we find \(F_1\) and \(F_2\):

\[ F_1=-\frac{1}{d\sigma_A}\left(-M^2\varphi_0+\sigma_{01}N^2\varphi_1\right), \]

\[ F_2=\frac{MN}{a_1d\sigma_A}\left(-a_2^2\sigma_{01}\varphi_0+a_1^2\varphi_1\right). \tag{12} \]

Substituting them into the penultimate equation (11), we obtain that

\[ \varphi_0=\frac{1}{a_2M}(a_1N\varphi_1-d\varphi_2). \tag{13} \]

Thus, if \(\varphi_3 \equiv \varphi_2\) and condition (13) is satisfied for \(\Delta_F\ne0\), system (1) has a solution in finite form (4), where \(A\) is determined by formula (5), \(B\) by formula (9), and \(F_1\) and \(F_2\) by formulas (12).

If, however, \(\Delta_F=0\), i.e. \(d=0\), and

\[ \frac{\varphi_0}{\varphi_1}=\frac{a_1^2}{a_2^2\sigma_{01}}, \]

then, for \(\varphi_3\equiv\varphi_2\), the case of I. A. Lappo-Danilevskii will occur, when in a system of the form

\[ \frac{dZ}{dt}=Z\cdot P \]

\(P\) commutes with its integral.

Indeed, from \(d=0\) we have

\[ a_2^2\sigma_{01}=\frac{a_1^2\sigma_0}{\sigma_1}. \]

Then

\[ \frac{\varphi_0}{\varphi_1}=\frac{\sigma_1}{\sigma_0}, \]

and system (1) will take the form

\[ \frac{dX}{dt} = X\left[(\sigma_1U_0+\sigma_0U_1)\frac{1}{\sigma_0}\varphi_1 +(U_0U_1+U_1U_0)\varphi_2\right], \]

where, according to the condition \(U_k^2=\sigma_kU_k\) \((k=0,1)\), the matrices \(\sigma_1U_0+\sigma_0U_1\) and \(U_0U_1+U_1U_0\) commute. The integral matrix in this case has the form

\[ X=\exp\left[ (\sigma_1U_0+\sigma_0U_1)\frac{1}{\sigma_0}\int_{t_0}^{t}\varphi_1\,d\tau + (U_0U_1+U_1U_0)\int_{t_0}^{t}\varphi_2\,d\tau \right]. \]

Let us now consider a system of the following form

\[ \frac{dX}{dt}=X(U_0\varphi_0+U_1\varphi_1+U_2\varphi_2), \tag{14} \]

where \(U_i\) \((i=0,1,2)\) are matrices of the second order with \(\det U_i=0\) \((i=0,1,2)\); \(\varphi_i\) \((i=0,1,2)\) are continuous scalar functions of \(t\). Using identity (10) § 1 in [2] and applying the substitution

\[ X=Y\exp\left[\left(\sigma_2-\frac{\sigma_{012}+\sigma_{102}}{\sigma_{01}}\right)\int_{t_0}^{t}\varphi_2\,d\tau\right], \]

if

\[ \sigma_2\ne \frac{\sigma_{012}+\sigma_{102}}{\sigma_{01}}, \]

we transform system (14) to the form

\[ \frac{dY}{dt} = Y\left[ U_0(\varphi_0+\sigma'\varphi_2) + U_1(\varphi_1+\sigma''\varphi_2) + \frac{\sigma_{012}U_0U_1+\sigma_{102}U_1U_0}{\sigma}\varphi_2 \right], \tag{14_1} \]

where

\[ \sigma'=\frac{\sigma_{01}\sigma_{12}-\sigma_1(\sigma_{012}+\sigma_{102})}{\sigma}, \qquad \sigma''=\frac{\sigma_{01}\sigma_{02}-\sigma_0(\sigma_{012}+\sigma_{102})}{\sigma}, \]

and

\[ \sigma=\sigma_{01}(\sigma_2-\sigma_0\sigma_1)\ne0, \]

while \(\sigma_{02},\sigma_{12},\sigma_{012}\) and \(\sigma_{102}\) are the traces of the corresponding matrices \(U_0U_2,\ U_1U_2,\ U_0U_1U_2\), and \(U_1U_0U_2\). In order that this system be reducible to system (10), it is necessary and sufficient that the equalities

\[ a_1F_1-\frac{\sigma_{01}N}{M}F_2=\varphi_0+\sigma'\varphi_2, \qquad \frac{a_2^2\sigma_{01}}{a_1}F_1-\frac{M}{N}F_2=\varphi_1+\sigma''\varphi_2, \tag{15} \]

\[ a_2 F_1+F_2=\frac{\sigma_{012}}{\sigma}\varphi_2,\qquad \sigma_{012}=\sigma_{102}. \tag{15} \]

Having found \(F_1\) and \(F_2\) from the first two equations under the condition that \(\Delta_F\ne 0\),

\[ F_1=\frac{1}{d\sigma_A}\left[-M^2\varphi_0+\sigma_{01}N^2\varphi_1+(\sigma_{01}\sigma''N^2-\sigma'M^2)\varphi_2\right], \]

\[ F_2=\frac{MN}{a_1d\sigma_A}\left[-a_2^2\sigma_{01}\varphi_0+a_1^2\varphi_1+(a_1^2\sigma''-a_2^2\sigma_{01}\sigma')\varphi_2\right], \tag{16} \]

and substituting them into the third equation of system (15), we obtain

\[ \varphi_0=\frac{1}{a_2M}\left[a_1N\varphi_1+\left(a_1\sigma''N-a_2\sigma'M-\frac{d\sigma_{012}}{\sigma}\right)\varphi_2\right]. \tag{17} \]

Thus, if for system (14) the conditions \(\sigma_{012}=\sigma_{102}\) and (17) are satisfied with \(\Delta_F\ne 0\), then the solution has the form (4), where \(F_1\) and \(F_2\) are determined by formulas (16).

If, however, \(\Delta_F=0\), i.e. \(d=0\), and

\[ \frac{\varphi_0+\sigma'\varphi_2}{\varphi_1+\sigma''\varphi_2} = \frac{a_1^2}{a_2^2\sigma_{01}}, \]

then, for \(\sigma_{012}=\sigma_{102}\), in system \((14_1)\) the case of I. A. Lappo-Danilevsky will occur.

Indeed, from \(d=0\) we have

\[ \frac{\varphi_0+\sigma'\varphi_2}{\varphi_1+\sigma''\varphi_2} = \frac{\sigma_1}{\sigma_0} \quad\text{or}\quad \varphi_0=\frac{\sigma_1}{\sigma_0}\varphi_1+ \left(\frac{\sigma_1\sigma''}{\sigma_0}-\sigma'\right)\varphi_2. \]

Then system \((14_1)\) can be written as

\[ \frac{dY}{dt} = Y\left\{ (\sigma_1U_0+\sigma_0U_1)\frac{1}{\sigma_0}\varphi_1+ \left[ (\sigma_1U_0+\sigma_0U_1)\frac{\sigma''}{\sigma_0} + (U_0U_1+U_1U_0)\frac{\sigma_{012}}{\sigma} \right]\varphi_2 \right\}. \]

The integral matrix of this system has the form

\[ Y=\exp\left\{ (\sigma_1U_0+\sigma_0U_1)\frac{1}{\sigma_0} \int_{t_0}^{t}\varphi_1\,d\tau + \left[ (\sigma_1U_0+\sigma_0U_1)\frac{\sigma''}{\sigma_0} + (U_0U_1+U_1U_0)\frac{\sigma_{012}}{\sigma} \right] \int_{t_0}^{t}\varphi_2\,d\tau \right\}, \]

since the matrices \(\sigma_1U_0+\sigma_0U_1\) and \(U_0U_1+U_1U_0\) commute.

Remark. From the results obtained it is easy to see that all conclusions generalize to any number of differential substitutions.

Literature

1. Erugin N. P. Linear systems of ordinary differential equations, Minsk, 1963.
2. Fedorov G. F. Izv. vuzov, Mathematics, No. 3, 217—224, 1958.
3. Lappo-Danilevsky I. A. Application of functions of matrices to the theory of linear systems of ordinary differential equations. GITTL, 1957.

Received by the editors
January 29, 1965

Institute of Mathematics, Academy of Sciences of the BSSR