Toronto Math Forum
APM3462012 => APM346 Math => Final Exam => Topic started by: Djirar on December 20, 2012, 01:30:05 PM

Solve the firstorder equation $2 u_t + 3 u_x =0$ with the auxiliary condition $u = \sin x$ when $t=0$.

I thought we were supposed to wait until Prof. Ivrii posted the problems...?
In any case, my solution to Problem 1 is attached.

I'm adding my solution.. the others do not mention why the solution works for all of $(x, t)$ (though it does not change the final result).

My answer to question 1

This was not just easy, it was Dan Quayle easy, so my grading was easy.
The easiest solution: Step 1. equation of characteristics:
$$
\frac{dx}{3}=\frac{dt}{2}\implies x\frac{3}{2}t =C \implies u=f(x\frac{3}{2}t)
$$
is a general solution. Other equivalent forms are possible leading to the same final answer, but this one is the most natural and straightforward.
Step 2 Initial condition: $u(x,0)=f(x)=\sin(x)$ and therefore
$$
\boxed{u(x,t)=\sin(x\frac{3}{2}t)}.
$$
Several students put the wrong sign $u=\sin(x+\frac{3}{2}t)$, several made mistakes on Step 2 and got marks halved. Few made really grave mistakes like trying method of separation, but majority did well and got all 20 (correct but ugly solutions/answers are not punished).