Example 1. Solve the Fredholm integral equation of the first kind

$$
\int_{0}^{1} K(x, t) \varphi(t) d t=\sin ^{3} \pi x
$$

where

$$
K(x, t)= \begin{cases}(1-x) t, & 0 \leqslant t \leqslant x \\ x(1-t), & x \leqslant t \leqslant 1\end{cases}
$$
Solution. The characteristic numbers of the kernel (7)

$$
\lambda_{1}=\pi^{2}, \quad \lambda_{2}=(2 \pi)^{2}, \quad \ldots, \quad \lambda_{n}=(n \pi)^{2}, \quad \ldots
$$

and the corresponding eigenfunctions

$$
\varphi_{1}(x)=\sqrt{2} \sin \pi x, \quad \varphi_{2}(x)=\sqrt{2} \sin 2 \pi x, \quad \ldots, \quad \varphi_{n}(x)=\sqrt{2} \sin n \pi x
$$

To determine the solvability of equation (6) for the kernel $K(x, t)$ given by formula (7), we will use the Picard theorem. Condition $1^{\circ}$ of the Picard theorem is clearly satisfied: the kernel (7) is real and symmetric.

The right-hand side of equation (6) can be represented as

$$
\sin ^{3} \pi x=\frac{3}{4} \sin \pi x-\frac{1}{4} \sin 3 \pi x
$$

or, equivalently,

$$
\sin ^{3} \pi x=\frac{3}{4 \sqrt{2}}(\sqrt{2} \sin \pi x)-\frac{1}{4 \sqrt{2}}(\sqrt{2} \sin 3 \pi x)=\frac{3}{4 \sqrt{2}} \varphi_{1}(x)-\frac{1}{4 \sqrt{2}} \varphi_{3}(x),
$$

where $\varphi_{1}(x)$ and $\varphi_{3}(x)$ are eigenfunctions from the system (8). Thus,

$$
f(x)=\sin ^{3} \pi x
$$

has the following coefficients in the expansion over the functions of the system (8):

$$
f_{1}=\frac{3}{4 \sqrt{2}}, \quad f_{2}=0, \quad f_{3}=-\frac{1}{4 \sqrt{2}}, \quad f_{k}=0 \quad(k \geqslant 4)
$$

The series (3) in this case reduces to a finite sum

$$
\sum_{k=1}^{\infty} \lambda_{k}^{2} f_{k}^{2}=\left(\frac{3}{4 \sqrt{2}}\right)^{2}\left(\pi^{2}\right)^{2}+\left(-\frac{1}{4 \sqrt{2}}\right)^{2}\left(3^{2} \pi^{2}\right)^{2}=\frac{45}{16} \pi^{4}
$$

Thus, condition $2^{\circ}$ is also satisfied.

Finally, the system of eigenfunctions (8) is a complete orthonormal system on $[0,1]$, so condition $3^{\circ}$ is also satisfied.

According to the Picard theorem, equation (6) has a unique solution, which can be represented as (5):

$$
\varphi(x)=\lambda_{1} f_{1} \varphi_{1}(x)+\lambda_{3} f_{3} \varphi_{3}(x)
$$

which in our case gives

$$
\varphi(x)=\frac{3 \pi^{2}}{4}(\sin \pi x-3 \sin 3 \pi x) . \quad \triangleright
$$

By direct substitution, we verify that the solution (9) is correct.

The requirement of completeness of the system of eigenfunctions $\left\{\varphi_{k}(x)\right\}$ in the Picard theorem is essential. We will demonstrate this with the following example.