I am kinda confuse with the second part of my homework. I did the first part (3/a and 4/a) without any problem, but part b for both problems I don't get it at all. I try to plug the boundaries in the u(x,t) equation, but for some reason i don't know what to do right after that. If somebody can help me I will be so thankfully. The problem is attached in a picture. 
2026-04-24 00:26:49.1776990409
Boundary conditions
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A Dirichlet condition is the prescription of the values of the solution at the boundary. A zero Dirichlet condition means the solution needs to be $0$ there. So you are asked to find the $\theta$ so that for this choice of $\theta$, you have $u(0,t)= 0$ and $u(l,t)=0$. Practically speaking, you set $u(0,t)= 0$, with the function $u$ you have, and solve for $\theta$, same for $l$.
A Neumann condition is the prescription of the derivative of the solution at the boundary.
A zero Neumann condition means the derivative, with respect to $x$ in your case, of the solution needs to be $0$ there. So you are asked to find the $\theta$ so that for this choice of $\theta$, you have $u_x(0,t)= 0$ and $u_x(l,t)=0$.