I would like to solve the boundary value problem $\frac{d^2 y}{dx^2} = f(x)$ with initial conditions $$y(-1)=y(1)=0$$ My thoughts were to try to find the solution without the conditions, and then filling in this solution in the differential equation to find all the right constants. However, I'm stuck.
I thought the following: $$y(x) = A + Bx + \int_{x_0}^x \left(\int_{x_0}^\eta f(\xi)d\xi \right)d\eta$$
But how can I proceed?