(i) The function $y(x)$ satisfies the differential equation

$$y^{\prime \prime}+b y^{\prime}+c y=0, \quad 0<x<1,$$

where $b$ and $c$ are constants, with boundary conditions $y(0)=0, y^{\prime}(0)=1$. By integrating this equation or otherwise, show that $y$ must also satisfy the integral equation

$$y(x)=g(x)+\int_{0}^{1} K(x, t) y(t) d t$$

and find the functions $g(x)$ and $K(x, t)$.

(ii) Solve the integral equation

$$\varphi(x)=1+\lambda^{2} \int_{0}^{x}(x-t) \varphi(t) d t, \quad x>0, \quad \lambda \text { real }$$

by finding an ordinary differential equation satisfied by $\varphi(x)$ together with boundary conditions.

Now solve the integral equation by the method of successive approximations and show that the solutions are the same.