12 HW 12

1. Solve 7.26, by Laplace transforming the problem as given in time. This is a good way to practice back transform methods. Note that one factor in $\widehat u$ is a simpler function at a shifted value of coordinate $s$.

2. Solve 7.35 by Laplace transform in time. Clean up completely; only the given function may be in your answer, no Heaviside functions or other weird stuff. There is a minor error in the book’s answer.

3. Write the complete (Sturm-Liouville) eigenvalue problem for the eigenfunctions of 7.27. Find the eigenfunctions of that problem. Make very sure you do not miss one. Write a symbolic expression for the eigenfunctions in terms of an index, and identify the values that that index takes. You may want, or not want, to write one eigenfunction explicitly instead of as a term in the sum.

4. Continuing the previous homework, write $f=x$ and $g=1$ in terms of the eigenfunctions you found for the case $\ell=1$. Be very careful with one particular eigenfunction. Note that sometimes you need to write a term in a sum or sequence out separately from the others.

5. Continuing the previous homework, substitute $u(x,t)=\sum_nu_n(t)X_n(x)$ (plus the additional term, if any) into the PDE to convert it into an ordinary differential for each separate coefficient $u_n(t)$. Solve the ODE. Be very careful with one particular case. By writing the initial conditions in terms of the eigenfunctions, identify the integration constants. Write out a complete summary of the solution. Make sure to identify the values of your numbering index in each expression.