I have the following wave equation here:
$\frac{\partial u(x,t)}{\partial x^2} -\frac{1}{c^2}\frac{\partial^2u}{\partial t^2}=6$ with the boundary conditions as follows:
$u(0,t)=0$
$u(a,t)=0$
$u(x,0)=0$
$\frac{d(u(x,t)}{dt}|_{t=0}=0$
Okay fine. No problems. So I go ahead and write the total solution $u(x,t)$ as $u(x,t)= U_p(x,t)+u_g(x,t)$ where $U_p(x,t)$ is the particular solution and $u_g(x,t)$ is the homogeneous solution.
So for the particular solution:
$\frac{d^2(u(x))}{dx^2}=6$
If I integrate twice and apply the first two boundary conditions, I get:
$U_p(x)=3x^2-3ax$ which works with the given boundary conditions.
Then I go ahead and attempt to solve the homogeneous solution. So in other words, I get:
$\frac{\partial u(x,t)}{\partial x^2} -\frac{1}{c^2}\frac{\partial^2u}{\partial t^2}=0$
So using the fact that they must both equal some speration constant, I solve the spatial differential equation first and obtain:
$\frac{1}{x}$$\frac{d^2X}{dx^2} -\frac{1}{c^2T}\frac{\partial^2T}{\partial t^2}=0$
So I get:
$\frac{1}{x}$$\frac{d^2X}{dx^2} = \lambda $ and $\frac{1}{c^2T}\frac{\partial^2T}{\partial t^2}=\lambda $
The spatial problem for $\lambda <0$ gives me a non trivial eigenfunction of:
$X_n(x) = A\sin(\frac{n\pi x}{a})$
For the time dependent differential equation:
$\frac{\partial^2T}{\partial t^2} = \lambda c^2 T $
So this means that I get:
$T(t) = c_1 \cos(\frac{c n \pi t}{a}) + c_2 \sin(\frac{c n \pi t}{a})$
So if I multiply the product solutions, I get:
$U_g(x,t) = A_n\cos(\frac{c n \pi t}{a})\sin(\frac{n \pi x}{a}) + B_n\sin(\frac{c n \pi t}{a})\sin(\frac{n \pi x}{a})$
To get the full solution, I just sum it all up so...
$$U_g(x,t) = \sum_{n=1}^{\infty} A_n\cos(\frac{c n \pi t}{a})\sin(\frac{n \pi x}{a}) + B_n\sin(\frac{c n \pi t}{a})\sin(\frac{n \pi x}{a})$$
Now the problem is that I'm not really sure how to use the last two boundary conditions, from what I am seeing I try to use then and find the Fourier sine coefficients, I get that $A_n$ and $B_n$ are both $0$? That implies I only have a particular function and no homogeneous term. It doesn't feel right to me...
Can someone please help in in understanding how would I use the last two boundary conditions?