I've been posed the following question:
I'm trying to solve for the solution to the partial differential equation but cannot seem to get it right.
the question specifies using the "separation of variables method" to solve it
My current attempt includes:
$\theta (x,t) = \sum_{n=1}^{\infty} u_n (x,t)$
$ = \sum_{n=1}^{\infty} (B_n cos \lambda _n t + C_n sin \lambda t) sin (\frac {n \pi x}{2 \pi})$
I'm not quite sure if this is the correct approach or how to solve for the coefficients.
Any help with this question will be greatly appreciated!
This equation can be solved analytically using for instance, the Laplace transform. You can use it as a means to certify your variable separation method.
In large steps, applying the Laplace transform
$$ s^2\Theta(x,s)-\theta(x,0)-s\theta_t(x,0) = \Theta_{xx}(x,s) $$
or
$$ s^2\Theta(x,s)-6\cos(3x)-s(5+4\cos(4x)) = \Theta_{xx}(x,s) $$
Solving for $x$
$$ \Theta(x,t) = c_1(s) e^{\frac{s x}{3}}+c_2(s) e^{-\frac{s x}{3}}+\frac{4 s^4 \cos (4 x)+5 s^4+6 s^3 \cos (3 x)+324 s^2 \cos (4 x)+1125 s^2+864 s \cos (3 x)+58320}{s \left(s^4+225 s^2+11664\right)} $$
and now we have
$$ \Theta_x(0,s) = \Theta_x(2\pi,s) $$
giving
$$ c_1(s)-c_2(s)=e^{\frac{2 \pi s}{3}} c_1(s)-e^{-\frac{2 \pi s}{3}} c_2(s) $$
or
$$ c_1(s) = -e^{-\frac{2 \pi s}{3}} c_2(s) $$
and finally
$$ \Theta(x,s) = c_2(s) \left(e^{-\frac{s x}{3}}-e^{\frac{1}{3} s (x-2 \pi )}\right)+\frac{6 \cos (3 x)}{s^2+81}+\frac{4 s \cos (4 x)}{s^2+144}+\frac{5}{s} $$
Here $c_2(s)$ represents the Laplace transform of any $f(t)$. Assuming now $\theta(0,t) = \sin(t)$ we have $c_2(s) = \frac{1}{s^2+1}$ and then
$$ \Theta(x,s) = \frac{e^{-\frac{s x}{3}}-e^{\frac{1}{3} s (x-2 \pi )}}{s^2+1}+\frac{6 \cos (3 x)}{s^2+81}+\frac{4 s \cos (4 x)}{s^2+144}+\frac{5}{s} $$
and anti-transforming
$$ \theta(x,t) = u \left(t-\frac{x}{3}\right) \sin \left(t-\frac{x}{3}\right)-u \left(t+\frac{1}{3} (x-2 \pi )\right) \sin \left(t+\frac{1}{3} (x-2 \pi )\right)+4 \cos (12 t) \cos (4 x)+\frac{2}{3} \sin (9 t) \cos (3 x)+5 $$
where $u(\cdot)$ is the Heaviside step function.
For $f(t) = 0$ as requested, then
$$ \theta(x,t) = 4 \cos (12 t) \cos (4 x)+\frac{2}{3} \sin (9 t) \cos (3 x)+5 $$
NOTE
For the problem
$$ \theta_{tt}=c^2\theta_{xx},\ \ \theta(0,t) = \theta(L,t) = 0, \ \ \theta(x,0) = \phi(x), \theta_t(x,0) = \psi(x) $$
the variables separation method says that
$$ \theta(x,t) = \sum_{n}^{\infty}\left(a_n\cos\left(\frac{n\pi c t}{L}\right)+b_n\sin\left(\frac{n\pi c t}{L}\right)\right)\sin\left(\frac{n\pi x}{L}\right) $$
with
$$ a_n = \frac 2L\int_0^L\phi(x)\sin\left(\frac{n\pi x}{L}\right)dx $$
and
$$ b_n = \frac {2}{n\pi c}\int_0^L\psi(x)\sin\left(\frac{n\pi x}{L}\right)dx $$
Here $L = 2\pi, \ \ c = 3, \ \ \phi(x) = 6\cos(3x),\ \ \psi(x) = 5+4\cos(4x)$
giving
$$ a_n = \frac{12 n (1-\cos (\pi n))}{\pi \left(n^2-36\right)}\\ b_n = \frac{12 \left(n^2-20\right) (1-\cos (\pi n))}{\pi n^2 \left(n^2-36\right)} $$