# Nonlinear PDE

Solve $xu_t + uu_x = 0$ with $u(x, 0) = x.$(Hint: Change variables $x \rightarrow x^2$.)
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 $x \rightarrow y^2$ I get the right solution, but again it turns out the change of variables was not necessary).