Work area for Practice Final Exam questions
Question:
Can anyone fill in the blanks on the last problem (23) Professor Bell worked in class today? I follow up to the u(x,t) = 1/2(sin2x)(cos2t) + ...... Where did the 1/2(sin2x)(cos2t) come from?
Answer:
When we did the method of separation of variables to solve the string problem, we got solutions of the form
X_n(x) T_n(t).
Then we took linear combinations and realized what the coefficients had to be from plugging in the initial conditions.
The cos 2t term is the T(t) part of the solution that goes with sin 2x. (The B_n coefficients are zero because there is zero initial velocity.)
Question:
Can someone explain the purpose of the infinite sum 1/n^2 in problem 30? I understand how to use the Parseval's identity, but that last term in the problem statement is really confusing me.
Answer:
When you square the coefficients of the Fourier series, you get four times the sum of 1/n^2.
Question:
Has anyone had any success with Problem 15? I keep solving this on and getting the solution A. I know I'm not doing it correctly. Any hints?
Answer: Your e^-2t term should be e^-2(t-1).
(Recall L(u(t-a)f(t-a))=e^-as*F(s)
