Example 5. Solve the integral equation

$$
\varphi(x)=x+\int_{x}^{\infty} \mathrm{e}^{2(x-t)} \varphi(t) d t
$$
Solution. In this case, $f(x)=x, K(x)=e^{2 x}$. Therefore,

$$
F(p)=\frac{1}{p^{2}}, \quad \overline{\mathscr{K}}(-p)=\int_{0}^{\infty} e^{-2 x} e^{p x} d x=\frac{1}{2-p}, \quad \operatorname{Re} p>1$, which is related to the inclusion or exclusion in the solution of equation (16) of the solution of the corresponding homogeneous equation

$$
\varphi(x)=\int_{x}^{\infty} e^{2(x-t)} \varphi(t) d t
$$

Let's find the residues of the integrand at its poles:

$$
\underset{p=0}{\operatorname{res}}\left(\frac{p-2}{p^{2}(p-1)} e^{p x}\right)=2 x+1, \quad \underset{p=1}{\operatorname{res}}\left(\frac{p-2}{p^{2}(p-1)} e^{p x}\right)=-e^{x}
$$

Therefore, the solution of the integral equation (16) is $\varphi(x)=2 x+1+C e^{x}$ ( $C$ - an arbitrary constant).

## Problems for independent solving

Solve the integral equations:

263. $\varphi(x)=e^{-x}+\int_{x}^{\infty} \varphi(t) d t$. 264. $\varphi(x)=e^{-x}+\int_{x}^{\infty} e^{x-t} \varphi(t) d t$.
264. $\varphi(x)=\cos x+\int_{x}^{\infty} e^{x-t} \varphi(t) d t$. 266. $\varphi(x)=1+\int_{x}^{\infty} e^{\alpha(x-t)} \varphi(t) d t \quad(\alpha>0)$.

## $5^{\circ}$. Generalized multiplication theorem and some of its

 applications. Let$$
\varphi(x) \risingdotseq \Phi(p), \quad u(x, \tau) \risingdotseq U(p) e^{-\tau \varphi(p)}
$$

where $U(p)$ and $q(p)$ are analytic functions. Then

$$
\Phi(q(p)) U(p) \risingdotseq \int_{0}^{\infty} \varphi(\tau) u(x, \tau) d \tau
$$

This is the generalized multiplication theorem (Erfos theorem). If $u(x, \tau)=u(x-\tau)$, then $q(p) \equiv p$, and we get the usual multiplication theorem:

$$
\Phi(p) \cdot U(p) \risingdotseq \int_{0}^{\infty} \varphi(\tau) u(x-\tau) d \tau
$$

If $U(p)=\frac{1}{\sqrt{p}}, q(p)=\sqrt{p}$, then

$$
u(x, \tau)=\frac{1}{\sqrt{\pi x}} e^{-\tau^{2} /(4 x)}
$$

Therefore, if it is known that $\Phi(p) \risingdotseq \varphi(x)$, then by the Erfos theorem we find the original for $\frac{\Phi(\sqrt{p})}{\sqrt{p}}:$

$$
\frac{\Phi(\sqrt{p})}{\sqrt{p}} \risingdotseq \frac{1}{\sqrt{\pi x}} \int_{0}^{\infty} \varphi(\tau) e^{-\tau^{2} /(4 x)} d \tau
$$