General Solution for 2nd order inhomogenous DE

Question

I am new with this topic so please bear with me if there are a lot of mistakes or misunderstandings.

I have been tasked with finding the general solution to:

$$\frac{d^2y}{dx^2} = \alpha y- \beta x(x-L)$$

where $\alpha$, $\beta$ and L are all real constants. We are also told boundary conditions that $$y(0) = 0$$ and $$y(L) = 0$$

Now for the homogeneous case I solved $$r^2-\alpha = 0 $$ giving $$(r-\sqrt\alpha)(r+\sqrt\alpha)$$ and it follows that $$y_h = C_1e^{-\sqrt\alpha x} + C_2e^{\sqrt\alpha x}$$

Now for the particular case I used the method of undetermined coefficients and arrived at

$$y_p = \frac{\beta}{\alpha}(x^2 - Lx + \frac{2}{\alpha})$$

so for the general solution I have

$$y = C_1e^{-\sqrt\alpha x} + C_2e^{\sqrt\alpha x} + \frac{\beta}{\alpha}(x^2 - Lx + \frac{2}{\alpha})$$

Could someone please tell me if I am taking the right approach here, and if not what I should be doing differently? and also how do I go about finding $C_1$ and $C_2$?

Answer 1

You did a good job. I checked your particular solution and all seems correct to me.

For the constants,

use the initial conditions

$y(L)= 0$ and $y(0)=0$ $$ \begin{cases} y(0)=0 \\ y(L)=0 \end{cases} \implies \begin{cases} 0 = C_1 + C_2 \\ 0 = C_1e^{-\sqrt\alpha L} + C_2e^{\sqrt\alpha L} \end{cases} $$ Solve these two equations to get $C_1,C_2$

Answer 2

HINT: use for the particular case the ansatz $$y_P=a_1+a_2x+a_3x^2$$