APM346-2012 > Final Exam

Problem 1

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Djirar:
Solve the first-order equation $2 u_t + 3 u_x =0$ with the auxiliary condition $u = \sin x$ when $t=0$.

Chen Ge Qu:
I thought we were supposed to wait until Prof. Ivrii posted the problems...?

In any case, my solution to Problem 1 is attached.

Ian Kivlichan:
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).

Pei Zhou:
My answer to question 1

Victor Ivrii:
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).

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