Example. Solve the integral equation

$$
\int_{0}^{x}(x-t) \varphi(t) d t=x^{2}
$$
Solution. In this case, $\beta=1, \lambda=2$. Since $\lambda-\beta+k \neq 0$ ( $k=$ $0,1,2, \ldots, n)$, then by formula (13)

$$
\varphi(x)=\frac{\Gamma(3)}{\Gamma(2) \Gamma(1)} x^{2-1-1}=2
$$

$\Delta$

## Problems for Independent Solution

Solve the integral equations:

59. $\int_{0}^{x}(x-t)^{1 / 3} \varphi(t) d t=x^{4 / 3}-x^{2}$. 60. $\int_{0}^{x}(x-t)^{1 / 2} \varphi(t) d t=\pi x$.
60. $\int_{0}^{x}(x-t)^{1 / 4} \varphi(t) d t=x+x^{2}$.
61. $\int_{0}^{x}(x-t)^{2} \varphi(t) d t=x^{3}$.
62. $\frac{1}{2} \int_{0}^{x}(x-t)^{2} \varphi(t) d t=\cos x-1+\frac{x^{2}}{2}$.

Chapter 2

## Fredholm Integral Equations

## § 6. Fredholm Equations. Basic Concepts

A linear integral equation of the second kind of Fredholm is an equation of the form

$$
\varphi(x)-\lambda \int_{a}^{b} K(x, t) \varphi(t) d t=f(x)
$$

where $\varphi(x)$ is the unknown function, $K(x, t)$ and $f(x)$ are known functions, $x$ and $t$ are real variables varying in the interval $(a, b)$, and $\lambda$ is a numerical factor.

The function $K(x, t)$ is called the kernel of the integral equation (1); it is assumed that the kernel $K(x, t)$ is defined in the square $\Omega\{a \leqslant x \leqslant b, a \leqslant t \leqslant b\}$ on the plane $(x, t)$ and is continuous in $\Omega$, or its discontinuities are such that the double integral

$$
\int_{a}^{b} \int_{a}^{b}|K(x, t)|^{2} d x d t
$$

has a finite value.

If $f(x) \not \equiv 0$, then equation (1) is called non-homogeneous; if $f(x) \equiv 0$, then equation (1) takes the form

$$
\varphi(x)-\lambda \int_{a}^{b} K(x, t) \varphi(t) d t=0
$$

and is called homogeneous.

An integral equation of the form

$$
\int_{a}^{b} K(x, t) \varphi(t) d t=f(x)
$$

which does not contain the sought function $\varphi(x)$ outside the integral, is called an integral equation of the first kind of Fredholm.

The limits of integration $a$ and $b$ in equations (1), (2), and (3) can be either finite or infinite.

A solution to the integral equations (1), (2), and (3) is any function $\varphi(x)$, the substitution of which into the equations turns them into identities for $x \in(a, b)$.