Example 1. Solve the integral equation

$$
\varphi(x)=\sin x+2 \int_{0}^{x} \cos (x-t) \varphi(t) d t
$$
Solution. It is known that

$$
\sin x \risingdotseq \frac{1}{p^{2}+1}, \quad \cos x \risingdotseq \frac{p}{p^{2}+1}
$$

Let $\varphi(x) \risingdotseq \Phi(p)$. Applying the Laplace transform to both sides of the equation and using the convolution theorem, we get

$$
\Phi(p)=\frac{1}{p^{2}+1}+\frac{2 p}{p^{2}+1} \Phi(p)
$$

From this,

$$
\Phi(p)\left[1-\frac{2 p}{p^{2}+1}\right]=\frac{1}{p^{2}+1}
$$

or

$$
\Phi(p)=\frac{1}{(p-1)^{2}} \risingdotseq x e^{x}
$$

Therefore, the solution to the integral equation is

$$
\varphi(x)=x e^{x}
$$

## Problems for Independent Solution

Solve the following integral equations:

234. $\varphi(x)=e^{x}-\int_{0}^{x} e^{x-t} \varphi(t) d t . \quad$ 235. $\varphi(x)=x-\int_{0}^{x} e^{x-t} \varphi(t) d t$.

![](https://cdn.mathpix.com/cropped/2024_05_22_d528bb4cd8a42024c50cg-114.jpg?height=125&width=1038&top_left_y=198&top_left_x=47)

238. $\varphi(x)=\cos x-\int_{0}^{x}(x-t) \cos (x-t) \varphi(t) d t$.
239. $\varphi(x)=1+x+\int_{0}^{x} e^{-2(x-t)} \varphi(t) d t . \quad 240 . \varphi(x)=x+\int_{0}^{x} \sin (x-t) \varphi(t) d t$.
240. $\varphi(x)=\sin x+\int_{0}^{x}(x-t) \varphi(t) d t . \quad$ 242. $\varphi(x)=x-\int_{0}^{x} \operatorname{sh}(x-t) \varphi(t) d t$.
241. $\varphi(x)=1-2 x-4 x^{2}+\int_{0}^{x}\left[3+6(x-t)-4(x-t)^{2}\right] \varphi(t) d t$.
242. $\varphi(x)=\operatorname{sh} x-\int_{0}^{x} \operatorname{ch}(x-t) \varphi(t) d t$. 245. $\varphi(x)=1+2 \int_{0}^{x} \cos (x-t) \varphi(t) d t$.
243. $\varphi(x)=e^{x}+2 \int_{0}^{x} \cos (x-t) \varphi(t) d t$. 247. $\varphi(x)=\cos x+\int_{0}^{x} \varphi(t) d t$.

The convolution theorem can also be used to solve nonlinear Volterra integral equations of the form

$$
\varphi(x)=f(x)+\lambda \int_{0}^{x} \varphi(t) \varphi(x-t) d t
$$

Let

$$
\varphi(x) \risingdotseq \Phi(p), \quad f(x) \rightleftharpoons F(p) .
$$

Then, by equation (3),

$$
\Phi(p)=F(p)+\lambda \Phi^{2}(p)
$$

from which

$$
\Phi(p)=\frac{1 \pm \sqrt{1-4 \lambda F(p)}}{2 \lambda}
$$

The original function for $\Phi(p)$, if it exists, will be the solution to the integral equation (3).