Toronto Math Forum

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

Title: Problem 1
Post by: Djirar on December 20, 2012, 01:30:05 PM
Solve the first-order equation $2 u_t + 3 u_x =0$ with the auxiliary condition $u = \sin x$ when $t=0$.

Title: Re: Problem 1
Post by: Chen Ge Qu on December 20, 2012, 01:35:12 PM
I thought we were supposed to wait until Prof. Ivrii posted the problems...?

In any case, my solution to Problem 1 is attached.
Title: Re: Problem 1
Post by: Ian Kivlichan on December 20, 2012, 02:20:06 PM
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).
Title: Re: Problem 1
Post by: Pei Zhou on December 20, 2012, 04:49:02 PM
Title: Re: Problem 1
Post by: Victor Ivrii on December 22, 2012, 07:40:22 AM
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).