# Thread: Help with Method of Separation of Variables

1. ## Help with Method of Separation of Variables

I need help with the following questions:

1.

The differential equation governing the displacement of a stretched string is [see attached equation].

Find the displacement of the string at any subsequent time t if the string is initially displaced as illustrated below and if it starts from rest.

[See attached graph]

Any help is most appreciated.

2. You fail to mention how much the string is initially displaced by at $\displaystyle (x,t) = \left( \frac{L}{2} , 0 \right)$. So let us call that value $\displaystyle M$.

The partial differencial equation is the 1d wave equation:
$\displaystyle \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial u^2}{\partial x^2}$.

The initial value problem is:
$\displaystyle \left\{ \begin{array}{c}u(x,0) = f(x) \\ u_t(x,0) = 0 \end{array}\right. \mbox{ for }0<x<L$.

Where $\displaystyle f(x)$ is the function described in your curve, which is,
$\displaystyle f(x) = \left\{ \begin{array}{c} \frac{2M}{L}x \mbox{ for }0 < x \leq \frac{L}{2} \\ \frac{2M}{L}\left( L - x \right) \mbox{ for }\frac{L}{2} \leq x < L \end{array} \right.$.

And the boundary value problem is:
$\displaystyle u(0,t) = u(L,t) = 0 \mbox{ for }t\geq 0$.

See if you can solve this wave equation from heir.