Function for a Nonhomogeous BVP

Question

What function of $\alpha$ satisfies the following conditions?

$ \alpha''(x)=x, $

$ \alpha(0)=0, $

$ \alpha(\pi)=0 $

Note the function is going to be used for a nonhomogeneous boundary value problem.

Answer 1

From your equation:

$$\alpha'(x) = \frac{1}{2} x^2 + C_1$$

$$\alpha(x) = \frac{1}{6} x^3 + C_1 x + C_2$$

The boundary conditions imply that $C_2=0$ and

$$\frac{\pi^3}{6} + C_1 \pi = 0 \implies C_1=-\frac{\pi^2}{6}$$

Then

$$\alpha(x) = \frac{1}{6} x^3 - \frac{\pi^2}{6} x$$