Problem2

[Aida Razi]

Solution is attached,

[Thomas Nutz]

I obtain the solution
$$
u(x,t)=\frac{1}{2}((x-ct)e^{x+ct}+(x-ct)e^{x-ct})+
\int_{x-ct}^{x+ct}e^{-x}dx+\int_0^{t}\int_{x-ct}^{x+ct}e^{-x-t}dxdt
$$
which becomes
$$
\frac{2ce^{-x}}{1-c^2}+\frac{e^{-t}(e^{-ct}+e^{ct})}{1+c}
$$
(plugging in $c=5$ doesn't really simplify it)

Did anyone get the same?

[Victor Ivrii]

Thomas,

I fixed you LaTeX (it was a double subscript error \int_{0{}_{t} but your formula is wrong: you should integrate by $x$ (inner) from $x-c(t-s)$ to $x+c(t-s)$ and integrand is $e^{x-s}$ and then you integrate by $s$ from $0$ to $t$; also forgot $\frac{1}{2c}$ factor.

I have not checked other terms. You can see that your "solution" does not satisfy initial conditions (or equations)

[Betty Zhu]

Razi, I agree with you, I got the same answer....hope it is the correct one...