Example 6. Solve the integral equation

$$
\frac{1}{\sqrt{\pi x}} \int_{0}^{\infty} e^{-t^{2} /(4 x)} \varphi(t) d t=1
$$
Solution. Let $\varphi(x) \risingdotseq \Phi(p)$. Applying the Laplace transform to both sides of (19), we get, according to formula (18),

$$
\frac{\Phi(\sqrt{p})}{\sqrt{p}}=\frac{1}{p}
$$

from which

$$
\frac{\Phi(p)}{p}=\frac{1}{p^{2}}, \quad \text { or } \quad \Phi(p)=\frac{1}{p} \risingdotseq 1
$$

Therefore, $\varphi(x) \equiv 1$ is a solution to equation (19).

## Problems for Independent Solution

Solve the following integral equations:

267. $\frac{1}{\sqrt{\pi x}} \int_{0}^{\infty} e^{-t^{2} /(4 x)} \varphi(t) d t=e^{-x} . \quad 268 . \frac{1}{\sqrt{\pi x}} \int_{0}^{\infty} e^{-t^{2} /(4 x)} \varphi(t) d t=2 x-\operatorname{sh} x$.
268. $\frac{1}{\sqrt{\pi x}} \int_{0}^{\infty} \mathrm{e}^{-t^{2} /(4 x)} \varphi(t) d t=x^{3 / 2}+e^{4 x}$.

It is known that

$$
t^{n / 2} J_{n}(2 \sqrt{t}) \risingdotseq \frac{1}{p^{n+1}} e^{-1 / p} \quad(n=0,1,2, \ldots)
$$

where $J_{n}(z)$ is the Bessel function of the first kind of order $n$. In particular,

$$
J_{0}(2 \sqrt{t}) \risingdotseq \frac{1}{p} e^{-1 / p}
$$

By the similarity theorem,

$$
J_{0}(2 \sqrt{x t}) \risingdotseq \frac{1}{p} e^{-x / p}
$$

from which it follows that for the Efros theorem, in this case, we should take

$$
q(p) \equiv \frac{1}{p}
$$