Example 3. Using the resolvent, find the solution to the integral equation

$$
\varphi(x)=e^{x^{2}}+\int_{0}^{x} e^{x^{2}-t^{2}} \varphi(t) d t
$$
Solution. The resolvent of the kernel $K(x, t)=\mathrm{e}^{x^{2}-t^{2}}$ for $\lambda=1$ is $R(x, t ; 1)=$ $e^{x-t} e^{\frac{x^{2}-t^{2}}{}}$ (see problem 26). According to formula (7), the solution of the given integral equation is the function

$$
\varphi(x)=e^{x^{2}}+\int_{0}^{x} e^{x-t} e^{x^{2}-t^{2}} e^{t^{2}} d t=e^{x+x^{2}}
$$

## Problems for Independent Solution

Using the results of the previous problems, find the solutions of the following integral equations using the resolvent:

36. $\varphi(x)=\mathrm{e}^{x}+\int_{0}^{x} e^{x-t} \varphi(t) d t$. 37. $\varphi(x)=\sin x+2 \int_{0}^{x} e^{x-t} \varphi(t) d t$.
37. $\varphi(x)=x 3^{x}-\int_{0}^{x} 3^{x-t} \varphi(t) d t$. 39. $\varphi(x)=e^{x} \sin x+\int_{0}^{x} \frac{2+\cos x}{2+\cos t} \varphi(t) d t$.
38. $\varphi(x)=1-2 x-\int_{0}^{x} e^{x^{2}-t^{2}} \varphi(t) d t$.
39. $\varphi(x)=e^{x^{2}+2 x}+2 \int_{0}^{x} e^{x^{2}-t^{2}} \varphi(t) d t$.
40. $\varphi(x)=1+x^{2}+\int_{0}^{x} \frac{1+x^{2}}{1+t^{2}} \varphi(t) d t$.

Remark. The unique solvability of Volterra integral equations of the 2nd kind

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

holds under much more general assumptions about the function $f(x)$ and the kernel $K(x, t)$ than their continuity.

Theorem. The