Toronto Math Forum

For this problem we have the following equations: $u_{xx} + u_{yy} = 0 \space \space (-\infty < x < \infty , y > 0)$ and $u|_{y=0} = f(x)$.
After doing the Fourier Transformation, the equations become $\hat{u}_{yy} - k^2\hat{u} = 0$ and $\hat{u}(k,0) = \hat{f}(k)$.
In my understanding, we should get a general form as $\hat{u}(k,y) = A(k)e^{ky} + B(k)e^{-ky}$ and drop the first term if $k>0$ since as $y \rightarrow \infty$ the term $e^{ky} \rightarrow \infty$, and if $y<0$ we drop the second one.
On the answer provided by Prof. Kennedy it said $\hat{u}(k,y) = \hat{f}(k)e^{-k|y|}$, so I am a little confused since $\hat{u}(k,y) = \hat{f}(k)e^{-|k|y}$ makes more sense to me.

In chapter 4.2, example 6, I was a little confused about the notation. My understanding is that $\omega_n$ are the roots for equation $\tan(\omega l) = \frac{(\alpha + \beta)\omega}{\omega^2 - \alpha\beta}$, so should the equation for $X_n$ be $X_n = \omega_n \cos(\omega_n x) + \alpha\sin(\omega_n x)$ instead of $X_n = \omega \cos(\omega_n x) + \alpha\sin(\omega_n x)$ (the first $\omega$ change to $\omega_n$)? I attached a screenshot of the example.

Another minor thing that I noticed is that for homework assignment 6/7, it said problems 1-11 for section 4.1 and 4.2, but there are no problem 11 in neither textbook version, so I guess it was a typo on Quercus.

For week 5 homework assignment, problem 13 of chapter 3.2 is assigned, but there is no problem 13 in the link to the online textbook. I found problem 13 in the pdf version of the textbook, so doing problem 13 in the pdf version is the same?

Another minor typo I noticed is that in online textbook chapter 3.2, theorem 3, I think it should be $\varepsilon > 0$. I marked the error in the attached screenshot.

I was a little confused about the region marked in red in this example. For the region on the right we get $\phi(x) = \sin(3x)$ and $\psi(x) = 3\sin(3x)$,  and from the boundary conditions we can get $\phi(x) = \psi(x) = 0$ for $t > 0$, so we have $u = 0$ for the region on the left. For the middle region in red, since $\psi(x - \frac{t}{3})$ is undefined, we can just have it as $\psi(x) = 0$ instead?  The middle region is like a mix of the other two regions so I am a little uncertain.

Chapter 2.2 problem 2 equation (9): $u_t + yu_x + xu_y = x$
and we get $\frac{dt}{1} = \frac{dx}{y} = \frac{dy}{x} = \frac{du}{x}$
By solving $\frac{dx}{y} = \frac{dy}{x}$ first we get $C = x^2 - y^2$
But we solving for either $\frac{dt}{1} = \frac{dx}{y}$ or $\frac{dt}{1} = \frac{dy}{x}$ we need to substitute $x$ for $y$ or the other direction, so in this case we will have a square root function like $x = \pm \sqrt{y^2 + C}$
I am not entirely sure whether or not we should proceed with the $\pm$ sign here or is there another easier approach to this question?