- #1

evinda

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MHB

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I want to solve the equation $xu_t+uu_x=0$ with $u(x,0)=x$. There is a hint to change variables $x \mapsto x^2$.

I have tried the following.

Let $x=y^2$.

Then $u_x=\frac{du}{dy} \frac{dy}{dx}+\frac{du}{dt} \frac{dt}{dx}=u_y \frac{1}{\frac{dx}{dy}}=\frac{1}{2y} u_y$.

Thus, $xu_t+uu_x=0 \Leftrightarrow y^2 u_t+\frac{1}{2y}uu_y=0$. But the latter doesn't help us to use the method of my post [m]Solve equation[/m].

So do we have to make an other change of variables? (Thinking)