Solution of Differential equation as an integral equation I was looking for the solution of the following problem.
Prove that if $\phi$ is a solution of the integral equation
$$y(t) = e^{it} + \alpha \int\limits_{t}^\infty \sin(t-\xi)\frac{y(\xi)}{\xi^2}d\xi,$$
then $\phi$ satisfies the differential equation
$$y'' + (1+\frac{\alpha}{t^2})y=0$$ 
Do I need to solve the differential equation to get the integral equation or I have to solve the integral equation to get the differential equation.

Take the derivative of $y$ twice using the equation,
$$y(t) = e^{it} + \alpha\int\limits_{t}^\infty \sin(t-\xi)\frac{y(\xi)}{\xi^2}d\xi$$
Then plug what you get into
$$y''+(1+\frac{\alpha}{t^2})y$$
and simplify to get $0$ and thus show the two are equivalent.
