Example 7. Solve the system of equations

$$
\frac{d t}{4 y-5 x}=\frac{d x}{5 t-3 y}=\frac{d y}{3 x-4 t}
$$
Solution. Multiplying the numerators and denominators of the fractions in system (23) by $3, 4, 5$ respectively and adding the numerators and denominators, we obtain, by (19),

$$
\frac{3 d t}{12 y-15 x}=\frac{4 d x}{20 t-12 y}=\frac{5 d y}{15 x-20 t}=\frac{3 d t+4 d x+5 d y}{0}
$$

(here $\lambda_{1}=3, \lambda_{2}=4, \lambda_{3}=5$). From this, $3 d t+4 d x+5 d y$ or $d(3 t+4 x+5 y)=0$, which means $3 t+4 x+5 y=C_{1}$ - this is the first integral of system (23).

Multiplying the numerators and denominators of the fractions in system (23) by $\lambda_{1}=2 t, \lambda_{2}=2 x, \lambda_{3}=2 y$ respectively and adding the numerators and denominators, we obtain, by (19),

$$
\frac{2 t d t}{8 y t-10 x t}=\frac{2 x d x}{10 t x-6 y x}=\frac{2 y d y}{6 x y-8 t y}=\frac{2 t d t+2 x d x+2 y d y}{0}
$$

from which

$$
2 t d x+2 x d x+2 y d y=0, \quad \text { or } \quad d\left(t^{2}+x^{2}+y^{2}\right)=0,
$$

and, therefore, the second first integral will be $t^{2}+x^{2}+y^{2}=C_{2}$.

The combination of the first integrals, which are independent, gives the general integral of system (23):

$$
3 t+4 x+5 y=C_{1}, \quad t^{2}+x^{2}+y^{2}=C_{2} .
$$

Thus, system (23) is solved.

## Problems for Independent Solution

Solve the following systems of differential equations:

796. $\frac{d t}{t}=\frac{d x}{x}=\frac{d y}{t y}$.
797. $\frac{d x}{y}=-\frac{d y}{x}=\frac{d p}{q}=-\frac{d q}{p}$.
798. $\frac{d t}{t^{2}-x^{2}-y^{2}}=\frac{d x}{2 t x}=\frac{d y}{2 t y}$.
799. $\left\{\begin{array}{l}t d x=(t-2 x) d t, \\ t d y=(t x+t y+2 x-t) d t .\end{array}\right.$
800. $\frac{d t}{x y}=\frac{d x}{y t}=\frac{d y}{x t}$.
801. $\frac{d x}{x t}=\frac{d y}{-y t}=\frac{d t}{x y}$.
802. $\left\{\begin{array}{l}\frac{d x}{d t}=\frac{3 t-4 y}{2 y-3 x}, \\ \frac{d y}{d t}=\frac{4 x-2 t}{2 y-3 x} .\end{array}\right.$
803. $\frac{t d t}{x^{2}-2 x y-y^{2}}=\frac{d x}{x+y}=\frac{d y}{x-y}$.

## § 22. Integration of Homogeneous Linear Systems with Constant Coefficients. Euler's Method

A homogeneous linear system with constant coefficients is a system of differential equations of the form

$$
\frac{d x_{i}}{d t}=\sum_{k=1}^{n} a_{i k} x_{k}(t), \quad i=1,2, \ldots, n
$$

where the coefficients $a_{i k}$ are constants, and $x_{k}(t)$ are the unknown functions of $t$.

The system (1) can be briefly written as a single matrix equation

$$
\frac{d X}{d t}=A X
$$

where

$$
A=\left(\begin{array}{ccc}
a_{11} & a_{12} & a_{1 n} \\
a_{21} & a_{22} & a_{2 n} \\
\cdots \cdots & \ldots \ldots & \ldots \\
a_{n 1} & a_{n 2} & a_{n n}
\end{array}\right), \quad X=\left(\begin{array}{c}
x_{1}(t) \\
x_{2}(t) \\
\vdots \\
x_{n}(t)
\end{array}\right), \quad \frac{d X}{d t}=\left(\begin{array}{c}
\frac{d x_{1}}{d t} \\
\frac{d x_{2}}{d t} \\
\vdots \\
\frac{d x_{n}}{d t}
\end{array}\right)
$$

A column matrix

$$
Y(t)=\left(\begin{array}{c}
y_{1}(t) \\
y_{2}(t) \\
\vdots \\
y_{n}(t)
\end{array}\right)
$$

is called a particular solution of equation (2) in the interval $(a, b)$ if the identity

$$
\frac{d Y}{d t} \equiv A Y(t) \quad \text { for } \quad a<t<b
$$

holds.

A system of particular solutions

$$
X_{1}(t)=\left(\begin{array}{c}
x_{1}^{(1)}(t) \\
x_{1}^{(2)}(t) \\
\vdots \\
x_{1}^{(n)}(t)
\end{array}\right), \quad X_{2}(t)=\left(\begin{array}{c}
x_{2}^{(1)}(t) \\
x_{2}^{(2)}(t) \\
\vdots \\
x_{2}^{(n)}(t)
\end{array}\right), \quad \ldots, \quad X_{n}(t)=\left(\begin{array}{c}
x_{n}^{(1)}(t) \\
x_{n}^{(2)}(t) \\
\vdots \\
x_{n}^{(n)}(t)
\end{array}\right)
$$

(here in the notation $x_{i}^{k}$ the lower index indicates the number of the solution, and the upper index indicates the function number in the solution) is called fundamental on the interval $(a, b)$ if its Wronskian determinant

$$
W(t) \equiv W\left(X_{1}, X_{2}, \ldots X_{n}\right)=\left|\begin{array}{cccc}
x_{1}^{(1)}(t) & x_{2}^{(1)}(t) & & x_{n}^{(1)}(t) \\
x_{1}^{(2)}(t) & x_{2}^{(2)}(t) & \ldots & x_{n}^{(2)}(t) \\
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \\
x_{1}^{(n)}(t) & x_{2}^{(n)}(t) & & x_{n}^{(n)}(t)
\end{array}\right| \neq 0
$$

for all $t \in(a, b)$.

Theorem. If a system of particular solutions of the homogeneous equation (2) is fundamental, then the general solution of this equation has the form

$$
X(t)=C_{1} X_{1}(t)+C_{2} X_{2}(t)+\ldots+C_{n} X_{n}(t)
$$

where $C_{1}, C_{2}, \ldots, C_{n}$ are arbitrary constants.

Linear systems can be integrated by various methods discussed earlier, such as the method of elimination, finding integrable combinations, etc.

For integrating homogeneous linear systems with constant coefficients, Euler's method is also applied.

Consider this method applied to a system of three linear differential equations:

$$
\left\{\begin{array}{l}
\frac{d x}{d t}=a x+b y+c z \\
\frac{d y}{d t}=a_{1} x+b_{1} y+c_{1} z \\
\frac{d z}{d t}=a_{2} x+b_{2} y+c_{2} z
\end{array}\right.
$$

We seek the solution of system (3) in the form

