Example 1. Formulate the integral equation corresponding to the differential equation

$$
y^{\prime \prime}+x y^{\prime}+y=0
$$

and the initial conditions

$$
y(0)=1, \quad y^{\prime}(0)=0
$$
Solution. Suppose

$$
\frac{d^{2} y}{d x^{2}}=\varphi(x)
$$

Then

$$
\frac{d y}{d x}=\int_{0}^{x} \varphi(t) d t+y^{\prime}(0)=\int_{0}^{x} \varphi(t) d t, \quad y=\int_{0}^{x}(x-t) \varphi(t) d t+1
$$

Substituting (9) and (10) into the given differential equation, we find

$$
\varphi(x)+\int_{0}^{x} x \varphi(t) d t+\int_{0}^{x}(x-t) \varphi(t) d t+1=0
$$

or

$$
\varphi(x)=-1-\int_{0}^{x}(2 x-t) \varphi(t) d t
$$

## Problems for Independent Solution

Formulate the integral equations corresponding to the following differential equations with the given initial conditions:

9. $y^{\prime \prime}+y=0 ; \quad y(0)=0, y^{\prime}(0)=1$. \quad 10. $y^{\prime}-y=0 ; \quad y(0)=1$.
10. $y^{\prime \prime}+y=\cos x ; \quad y(0)=y^{\prime}(0)=0$.
11. $y^{\prime \prime}-5 y^{\prime}+6 y=0$;

$y(0)=0, y^{\prime}(0)=1$.

13. $y^{\prime \prime}+y=\cos x$

$y(0)=0, y^{\prime}(0)=1$.

14. $y^{\prime \prime}-y^{\prime} \sin x+\mathrm{e}^{x} y=x ; y(0)=1, y^{\prime}(0)=-1$.
15. $y^{\prime \prime}+\left(1+x^{2}\right) y=\cos x ; y(0)=0, y^{\prime}(0)=2$.
16. $y^{\prime \prime \prime}+x y^{\prime \prime}+\left(x^{2}-x\right) y=x \mathrm{e}^{x}+1 ; y(0)=y^{\prime}(0)=1, y^{\prime \prime}(0)=0$.
17. $y^{\prime \prime \prime}-2 x y=0 ; \quad y(0)=\frac{1}{2}, y^{\prime}(0)=y^{\prime \prime}(0)=1$.
18. Show that a linear differential equation with constant coefficients under any initial conditions can be reduced to an integral equation of the Volterra type of the second kind with a kernel depending only on the difference of the arguments $(x-t)$ (an integral equation with a closed cycle or a convolution equation).

Some particular forms of Volterra integral equations of the first and second kind can be solved by reducing them to differential equations.