## Example 1.

$$
\varphi(x)-\lambda \int_{0}^{1}\left(5 x^{2}-3\right) t^{2} \varphi(t) d t=e^{x}
$$
Solution. We have

$$
\varphi(x)=C \lambda\left(5 x^{2}-3\right)+e^{x}
$$

and

$$
C=\int_{0}^{1} t^{2} \varphi(t) d t
$$

Substituting (6) into (7), we get

$$
C=C \lambda \int_{0}^{1}\left(5 t^{4}-3 t^{2}\right) d t+\int_{0}^{1} t^{2} e^{t} d t
$$

from which

$$
C=e-2 \text {. }
$$

This equation has a unique solution for any $\lambda$

$$
\varphi(x)=\lambda(e-2)\left(5 x^{2}-3\right)+e^{x},
$$

and the corresponding homogeneous equation

$$
\varphi(x)-\lambda \int_{0}^{1}\left(5 x^{2}-3\right) t^{2} \varphi(t) d t=0
$$

has the unique zero solution: $\varphi(x) \equiv 0$.