Find a linear combination of $u_n$'s satisfying $u(x,1) = \sin(2\pi x) -\sin(3\pi x)$

Question

I have the following problem:

$$u_n(x,y) = \sin(n\pi x)\sinh(n\pi y), \;\;\;n = 1, 2, 3, ...$$

Find a linear combination of the $u_n$'s that satisfies:

$$u(x,1) = \sin(2\pi x) -\sin(3\pi x)$$

Any hints? =)

Answer 1

I think I got this, I don't know why it was so difficult to see, its very simple xD, I guess I thought it too difficult x)

Why can't I just do:

$$\frac{1}{\sinh(2\pi)}u_2(x,1) - \frac{1}{\sinh(3\pi)}u_3(x,1) = $$$$\frac{1}{\sinh(2\pi)}\sin(2\pi x)\sinh(2\pi) - \frac{1}{\sinh(3\pi)}\sin(3\pi x)\sinh(3\pi) = $$ $$\sin(2\pi x) - \sin(3\pi x)$$ =)

So $$u(x,1) = \frac{1}{\sinh(2\pi)}u_2(x,1) - \frac{1}{\sinh(3\pi)}u_3(x,1) = \sin(2\pi x) - \sin(3\pi x)$$