# Homework #3

## 12 thoughts on “Homework #3”

1. For question 2 when you get to the point of integrating the side with the sum and alpha_k how do we know whether k = m or not?

• I take it you’re ending up with something like: $\sum_{k=1}^\infty \alpha_k\int_0^L \left(e^{i\frac{k\pi}{L}x}-e^{-i\frac{k\pi}{L}x}\right)\left(\overline{e^{i\frac{m\pi}{L}x}-e^{-i\frac{m\pi}{L}x}}\right)\,dx$
The sum is shorthand for :
plug in k=1, calculate the integral, multiply by $\alpha_1$
plug in k=2, calculate the integral, multiply by $\alpha_2$
plug in k=3, …
(and add up all the results)

unless k=m, the integral is just going to give zero, so you the sum reduces to: $0 + 0 + \dots + 0 + \alpha_m\,\cdot 2L + 0 + 0 + \dots$

2. I am having trouble with matching and solving alpha_k. As in I am left with the integral of u(x,0) * conjugate complex exponentials = alpha_m * 2 * L. Do we manually solve these integrals? I am not able to match what is done in class as a solution.

• You’re on the right track. Usually you only need to solve the integral twice, once for $k=0$, and again for $k\neq 0$.

3. Question from a student:

In question 4a, If one of the constraint is a derivative of V with respect to x, I am not sure what to even do

This is a separation of variables question, so you’ve made the guess $v(x,t) = X(x)T(t)$. Plug this guess into the constraints.

For example, $v(2,t)=0$.
You plug in $X(2)T(t)=0$. You don’t want $T(t)=0$, because that makes $v(x,t)=0$, which is a solution, but not a helpful one. It must be then that $X(2)=0$.

Do a similar thing for $\partial_xv(0,t)=0$, then start solving the ODE for $X(x)$.

4. Question from a student:

In question 1a after I convert the sin + sin functions into exponential using Euler’s method, how do I proceed from there? I have looked at the slide mentioned but it only shows that if exponential with power of K’s multiplied by its conjugate(with power of M’s instead of K’s) is either 0 if K does not equal M and 2L if it is.

Try going the other way around, and convert slide 7 into sine functions.
You should get: \int_0^L\sin\left(\frac{k\pi}{L}x\right)\sin\left(\frac{m\pi}{L}x\right) = \left\{\begin{aligned}&0&&\text{ if }k\neq m\\&\frac{L}{2}&&\text{ if }k=m\end{aligned}\right.

5. For Question 4c, what exactly they mean by the general solution for part b?
The solution to the PDE with inhomogeneous constraints involves $u_p(x)$ and $u_h(x,t)$, and the latter requires to have f(x) from the data for the PDE, i.e. u(x,0) = f(x). How can we find the general solution if we are not given f(x)? Or does the question ask about some other solution?

• You’re on the right track. Without $f(x)$, you can still give a formula for $u_h(x,t)$ that includes unknown constants.

6. For the solution to 4b):

We had a constraint that Up(2) = -3, but it seems if Up(x) = x +1, then Up(2) = 2 + 1 = 3 (not -3).
Which is right? Just a little confused.
Thank you.