Example 2. Solve the integral equation:

$$
\int_{0}^{x} \varphi(t) \varphi(x-t) d t=\frac{x^{3}}{6}
$$
Solution. Let $\varphi(x) \risingdotseq \Phi(p)$. Applying the Laplace transform to both sides of (4), we get

$$
\Phi^{2}(p)=\frac{1}{p^{4}}
$$

from which

$$
\Phi(p)= \pm \frac{1}{p^{2}}
$$

The functions $\varphi_{1}(x)=x, \varphi_{2}(x)=-x$ are solutions to equation (4) (the solution to equation (4) is not unique).

## Problems for independent solving

Solve the integral equations:

248. $2 \varphi(x)-\int_{0}^{x} \varphi(t) \varphi(x-t) d t=\sin x$.
249. $\varphi(x)=\frac{1}{2} \int_{0}^{x} \varphi(t) \varphi(x-t) d t-\frac{1}{2} \operatorname{sh} x$.

## $2^{\circ}$. Systems of Volterra integral equations of convolution type. The Laplace transform can be used to solve systems of Volterra integral equations of the form

$$
\varphi_{i}(x)=f_{i}(x)+\sum_{j=1}^{s} \int_{0}^{x} K_{i j}(x-t) \varphi_{j}(t) d t \quad(i=1,2, \ldots, s)
$$

where $K_{i j}(x), f_{i}(x)$ are known continuous functions that have a Laplace transform.

Applying the Laplace transform to both sides of (5), we get

$$
\Phi_{i}(p)=F_{i}(p)+\sum_{j=1}^{b} \tilde{K}_{i j}(p) \Phi_{j}(p) \quad(i=1,2, \ldots, s)
$$

This is a system of linear algebraic equations with respect to $\Phi_{j}(p)$. Solving it, we find $\Phi_{j}(p)$, the originals of which will be the solutions to the original system of integral equations (5).