Example 3. Solve the integral equation

$$
\int_{-1}^{1} \frac{\varphi(t) d t}{\sqrt{1+x^{2}-2 x t}}=x+1
$$
## Solution. Function

$$
G(x, t)=\frac{1}{\sqrt{1+x^{2}-2 x t}}
$$

is the generating function for the Legendre polynomials $P_{n}(t)$:

$$
G(x, t)=\sum_{n=0}^{\infty} P_{n}(t) x^{n}
$$

We seek the solution of equation (16) in the form

$$
\varphi(x)=\sum_{i=0}^{\infty} a_{i} P_{i}(x)
$$

Substituting (17) and (18) into (16) and considering that

$$
\left\|P_{n}\right\|^{2}=\frac{2}{2 n+1}
$$

we will have

$$
\sum_{n=0}^{\infty} \frac{2 a_{n}}{2 n+1} x^{n}=x+1
$$

From this, $a_{0}=\frac{1}{2}, a_{1}=\frac{3}{2}, a_{n}=0(n \geqslant 2)$. Therefore, the solution of the given equation is

$$
\varphi(x)=\frac{1}{2} P_{0}(x)+\frac{3}{2} P_{1}(x)=\frac{1}{2}+\frac{3}{2} x
$$