Separation of Variables (PDE) for the Laplace Equation

[FAS1998]

TL;DR Summary

I've attached an image of a solved problem. Can somebody explain the steps in the yellow box? I don't understand how they got to that point from the previous steps.

 

[phyzguy]

I suspect it is the first step that is bothering you.
$$\sum_{n=1}^{\infty}a_n \sin(\frac{n \pi x}{2}) \sinh(\frac{n \pi}{2}) = \sin(\frac{\pi x}{2})$$
This is true for all values of x. The only way this can be true is for all of the $a_n$ to be zero except $a_1$. Is this the step that is troubling you? After this, the rest follows pretty easily.

 

[FAS1998]

I suspect it is the first step that is bothering you.
$$\sum_{n=1}^{\infty}a_n \sin(\frac{n \pi x}{2}) \sinh(\frac{n \pi}{2}) = \sin(\frac{\pi x}{2})$$
This is true for all values of x. The only way this can be true is for all of the $a_n$ to be zero except $a_1$. Is this the step that is troubling you? After this, the rest follows pretty easily.

That is the step that is bothering me. Can you explain why all values of a must be 0 except a1?

 

[phyzguy]

That is the step that is bothering me. Can you explain why all values of a must be 0 except a1?

The functions $\sin(\frac{n \pi x}{2})$ are orthogonal functions on the interval (0,2). If I integrate:
$$\int_0^2 \sin(\frac{n \pi x}{2}) \sin(\frac{m \pi x}{2}) dx = \delta_{nm}$$
so it is zero unless n=m. So if you take your original expression, multiply both sides by $\sin(\frac{m \pi x}{2})$ and integrate both sides you get:

$$\sum_{n=1}^{\infty}a_n \sinh(\frac{n \pi}{2}) \int_0^2 \sin(\frac{n \pi x}{2}) \sin(\frac{m \pi x}{2}) = \int_0^2 \sin(\frac{\pi x}{2}) \sin(\frac{m \pi x}{2}) dx$$

This gives $a_m \sinh(\frac{m \pi}{2}) = 0$ if m is not equal to 1, and $a_m \sinh(\frac{m \pi}{2}) = 1$ if m = 1.