APM346--2020S > Quiz 2

Quiz2 TUT5101

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Jingjing Cui:
$$2u_{t}+t^2u_{x}=0\\ \frac{dt}{2}=\frac{dx}{t^2}=\frac{du}{0}\\ \int\frac{1}{2}t^2dt=\int1dx\\ \frac{1}{6}t^3+A=x\\ A=x-\frac{1}{6}t^3\\$$
Because c=0, so
$$u(t,x)=g(A)=g(x-\frac{1}{6}t^3)$$

The initial condition given in the question: u(x,0)=f(x)
The characteristics curves ($A=x-\frac{1}{6}t^3$) will always intersect t=0 (x-axis) at a unique point, no matter what value A takes. Thus, the solution always exist.

Victor Ivrii:
In the tsecond/third lines should be $dt$, ..., not $\partial t$,...