Example 3. Solve the system of integral equations

$$
\left\{\begin{array}{l}
\varphi_{1}(x)=1-2 \int_{0}^{x} e^{2(x-t)} \varphi_{1}(t) d t+\int_{0}^{x} \varphi_{2}(t) d t \\
\varphi_{2}(x)=4 x-\int_{0}^{x} \varphi_{1}(t) d t+4 \int_{0}^{x}(x-t) \varphi_{2}(t) d t
\end{array}\right.
$$
Solution. Transitioning to images and using the convolution theorem, we obtain

$$
\left\{\begin{array}{l}
\Phi_{1}(p)=\frac{1}{p}-\frac{2}{p-2} \Phi_{1}(p)+\frac{1}{p} \Phi_{2}(p) \\
\Phi_{2}(p)=\frac{4}{p^{2}}-\frac{1}{p} \Phi_{1}(p)+\frac{4}{p^{2}} \Phi_{2}(p)
\end{array}\right.
$$

Solving the obtained system with respect to $\Phi_{1}(p)$ and $\Phi_{2}(p)$, we find

$$
\begin{aligned}
& \Phi_{1}(p)=\frac{p}{(p+1)^{2}}=\frac{1}{p+1}-\frac{1}{(p+1)^{2}} \\
& \Phi_{2}(p)=\frac{3 p+2}{(p-2)(p+1)^{2}}=\frac{8}{9} \cdot \frac{1}{p-2}+\frac{1}{3} \cdot \frac{1}{(p+1)^{2}}-\frac{8}{9} \cdot \frac{1}{p+1}
\end{aligned}
$$

The originals for $\Phi_{1}(p)$ and $\Phi_{2}(p)$ are respectively

$$
\begin{aligned}
\varphi_{1}(x) & =e^{-x}-x e^{-x} \\
\varphi_{2}(x) & =\frac{8}{9} e^{2 x}+\frac{1}{3} x e^{-x}-\frac{8}{9} e^{-x}
\end{aligned}
$$

The functions $\varphi_{1}(x), \varphi_{2}(x)$ provide the solution to the original system of integral equations (6).

## Problems for Independent Solution

Solve the following systems of integral equations:

250. 

$$
\left\{\begin{array}{l}
\varphi_{1}(x)=\sin x+\int_{0}^{x} \varphi_{2}(t) d t, \\
\varphi_{2}(x)=1-\cos x-\int_{0}^{x} \varphi_{1}(t) d t.
\end{array} \text { 251. } \left\{\begin{array}{l}
\varphi_{1}(x)=e^{2 x}+\int_{0}^{x} \varphi_{2}(t) d t \\
\varphi_{2}(x)=1-\int_{0}^{x} e^{2(x-t)} \varphi_{1}(t) d t
\end{array}\right.\right.
$$

$$
\begin{aligned}
& \left\{\begin{array}{l}
\varphi_{1}(x)=e^{x}+\int_{0}^{x} \varphi_{1}(t) d t-\int_{0}^{x} e^{x-t} \varphi_{2}(t) d t \\
\varphi_{2}(x)=-x-\int_{0}^{x}(x-t) \varphi_{1}(t) d t+\int_{0}^{x} \varphi_{2}(t) d t
\end{array}\right. \\
& \left\{\begin{array}{l}
\varphi_{1}(x)=e^{x}-\int_{0}^{x} \varphi_{1}(t) d t+4 \int_{0}^{x} e^{x-t} \varphi_{2}(t) d t \\
\varphi_{2}(x)=1-\int_{0}^{x} e^{t-x} \varphi_{1}(t) d t+\int_{0}^{x} \varphi_{2}(t) d t .
\end{array}\right. \\
& \varphi_{1}(x)=x+\int_{0}^{x} \varphi_{2}(t) d t \\
& \varphi_{2}(x)=1-\int_{0}^{x} \varphi_{1}(t) d t \\
& \varphi_{3}(x)=\sin x+\frac{1}{2} \int_{0}^{x}(x-t) \varphi_{1}(t) d t \\
& \text { 255. }\left\{\begin{array}{l}
\varphi_{1}(x)=1-\int_{0}^{x} \varphi_{2}(t) d t, \\
\varphi_{2}(x)=\cos x-1+\int_{0}^{x} \varphi_{3}(t) d t, \\
\varphi_{3}(x)=\cos x+\int_{0}^{x} \varphi_{1}(t) d t .
\end{array} \quad 256 . \quad\left\{\begin{array}{l}
\varphi_{1}(x)=x+1+\int_{0}^{x} \varphi_{3}(t) d t \\
\varphi_{2}(x)=-x+\int_{0}^{x}(x-t) \varphi_{1}(t) d t, \\
\varphi_{3}(x)=\cos x-1-\int_{0}^{x} \varphi_{1}(t) d t
\end{array}\right.\right.
\end{aligned}
$$

252. 

$3^{\circ}$. Integro-differential equations. Suppose we have a linear integro-differential equation of the form

$\varphi^{(n)}(x)+a_{1} \varphi^{(n-1)}(x)+\ldots+a_{n} \varphi(x)+\sum_{m=0}^{s} \int_{0}^{x} K_{m}(x-t) \varphi^{(m)}(t) d t=f(x)$,

where $a_{1}, a_{2}, \ldots, a_{n}$ are constants, $f(x), K_{m}(x)(m=0,1, \ldots, s)$ are known functions, and $\varphi(x)$ is the unknown function.

Initial conditions for the unknown function $\varphi(x)$ are given as

$$
\varphi(0)=\varphi_{0}, \quad \varphi^{\prime}(0)=\varphi_{0}^{\prime}, \quad \ldots, \quad \varphi^{(n-1)}(0)=\varphi_{0}^{(n-1)}
$$

Let the functions $f(x)$ and $K_{m}(x)$ be originals and

$$
f(x) \risingdotseq F(p), \quad K_{m}(x) \risingdotseq \widetilde{K}_{m}(p) \quad(m=0,1, \ldots, s)
$$

Then the function $\varphi(x)$ will have the Laplace transform $\varphi(x) \risingdotseq \Phi(p)$. Applying the Laplace transform to both sides of (7) and using the derivative image theorem and the multiplication theorem, we arrive at the equation

$$
\Phi(p)\left[p^{n}+a_{1} p^{n-1}+\ldots+a_{n}+\sum_{m=0}^{s} \widetilde{K}_{m}(p) p^{m}\right]=A(p)
$$

where $A(p)$ is some known function of $p$. From (9) we find $\Phi(p)$ - the operational solution to the problem (7)-(8). The function $\varphi(x) \risingdotseq \Phi(p)$ will be the solution to the integro-differential equation (7), satisfying the initial conditions (8).