
Nonlinear PDE
Okay so I am working on this problem:
Solve $\displaystyle xu_t + uu_x = 0$ with $\displaystyle u(x, 0) = x.$(Hint: Change variables $\displaystyle x \rightarrow x^2$.)
However, I am not sure how to use the change of variables hint that is given or why it is needed. My thinking is that I could just use the method of characteristic as normal to get the solution. Any help is appreciated.

Re: Nonlinear PDE
I have figured out the problem, it turns out that I was right and did not really need the hint (the method of characteristics worked fine). My confusion with the hint turned out to be that they used the same variable for the change of variables (they used x on both sides so when I plugged it in I ended up with the same equation where as if I did $\displaystyle x \rightarrow y^2 $ I get the right solution, but again it turns out the change of variables was not necessary).