Toronto Math Forum
APM3462019 => APM346Lectures & Home Assignments => Home Assignment 3 => Topic started by: Yilin Ye on January 30, 2019, 11:49:08 PM

Do we need to consider all situations when calculating
\begin{matrix}\frac{1}{2} \int_{xct}^{x+ct} h(x)\, dx\end{matrix}
Like
(1) xct,x+ct >1
(2)x+ct>1, xct<1
(3) 1<xct<x+ct<1
(4)1<x+ct<1,xct<1
(5)xct,x+ct<1

duplicate removed.
Yes, you need to consider all cases and it is recommended to draw a plane and different regions there

Dear professor,
do we always assume that xct < x+ct? or we have to take cases? thanks

Since in this problem nothing is said about $t>0$ the complete solution should cover all cases. However, since in the problems either $g(x)=0$, or $h(x)=0$ then $u(x,t)$ is odd or even with respect to $t$, respectively. At least in some problems you can observe that solution must be even or odd with respect to $x$ as well.
In such problem, as (17), we have several regions. But we need to work out only some of them and extend to the rest by above arguments.

Professor, may I ask how to derive all the five conditions mentioned above? Why it's not necessary to consider the condition like ${x+ct>1, xct<1}$ or ${xct>1, x+ct,1}$ something like this?

No, you draw lines according to conditions: $x=1+ct$, $x=1+ct$, $x=1ct$ and $x=1ct$.
And normally one need to consider all 9 domains. However, our problems have two symmetries and it is sufficient to consider only 4 domains intersecting with the 1st quadrant and extend solution to the remaining 5, using the fact that solution is either odd or even with respect to $t$ (you need to understand, what is the case), and also solution is either odd or even with respect to $x$ (you need to understand, what is the case).