I tried to resolve the following wave equation by separating variable: \begin{equation*} \frac{\partial^2 w(x,t)}{\partial x^2} = \frac{1}{a^2} \frac{\partial^2 w(x,t)}{\partial t^2}, \quad a > 0 \end{equation*}
with the following conditions: \begin{align*} \begin{cases} w(0, t) = w(L, t) = 0 \\ w(x, 0) = 3x^3 - 3L^2x \equiv f(x) \\ \frac{\partial w(x,t)}{\partial t} \Big|_{t=0} = 0 \end{cases} \end{align*}
We apply the separation of variables: \begin{equation*} w(x,t) = u(x)v(t) \end{equation*}
The EDP becomes: \begin{equation*} u''(x)v(t) = \frac{1}{a^2} u(x)v''t() \end{equation*}
We check the boundary conditions: \begin{align*} u(0)v(t) = u(L)v(t) = 0 \end{align*}
Obviously, $v(t) \neq 0$ otherwise $w(x,t) = 0$. Thus: \begin{align*} u(0) = u(L) = 0 \end{align*}
Let $K^2 = k^2 a^2$, we can solve the 2 EDOs: \begin{align*} \begin{cases} u''(x) = -k^2 u(x) \implies u(x) = A\cos(kx) + B\sin(kx) \\ v''(t) = - k^2 a^2 v(t) = - K^2 v(t) \implies v(t) = C\cos(Kt) + D\sin(Kt) \end{cases} \end{align*}
We rewrite the boundaries conditions: \begin{align*} u(0) &= 0 \implies A = 0 \\ u(L) &= 0 \implies B sin(kL) = 0 \implies kL = n\pi \quad; n \in \mathbb{Z} \end{align*}
We get: \begin{align*} u(x) &= B \sin(\frac{n\pi x}{L}) \\ v(t) &= C \sin(\frac{n^2 \pi^2 a^2 t}{L^2}) + D \cos(\frac{n^2 \pi^2 a^2 t}{L^2}) \end{align*} \begin{align*} w(x,t) = \sum_{n=1}^{+\infty} B \sin(\frac{n\pi x}{L}) \left[ C \sin(\frac{n^2 \pi^2 a^2 t}{L^2}) + D \cos(\frac{n^2 \pi^2 a^2 t}{L^2}) \right] \end{align*}
We apply the initial conditions: \begin{align*} w(x, 0) = 3x^3 - 3L^2x &= \sum_{n=1}^{+\infty} B \sin(\frac{n\pi x}{L}) D \\ &= \sum_{n=1}^{+\infty} E_n \sin(\frac{n\pi x}{L}) \end{align*}
We can find $E_n$ knowing that $f(x)$ is odd because $f(x) = -f(-x)$: \begin{align*} E_n = \frac{1}{L} \int_0^{L} (3x^3 - 3L^2x) \sin(\frac{n\pi x}{L}) dx \end{align*}
After multiple integrations by part, we find: \begin{align*} E_n = \left( \frac{18L^3}{n^3\pi^3} - \frac{15L^3}{n \pi} \right) (-1)^n \end{align*}
With the last initial condition, $F_n = B \cdot C = 0$ so the final solution is: \begin{align*} w(x,t) = \sum_{n=1}^{+\infty} \left( \frac{18L^3}{n^3\pi^3} - \frac{15L^3}{n \pi} \right) (-1)^n \sin(\frac{n\pi x}{L}) \cos(\frac{n^2 \pi^2 a^2 t}{L^2}) \end{align*}
Is it right ?
One last question that I do not really understand is "is the solution uniquely determined ?".
How can I check that ?
Firstly, you make a mistake when you calculate $E_n$ . The coefficient should be "$\frac{2}{L}$" instead of "$\frac{1}{L}$".
Secondly, $v(t)=Csin(\frac{n\pi at}{l})+Dcos(\frac{n\pi at}{l})$ (no square here!)
Since the general solution of $V''+K^{2}V=0$ is $Csinkt+Dcoskt$
As to the reason why the solution is unique, I'm sorry that I can't figure out a perfect proof....