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Consider the differential equation $$ u^2\frac{\partial^2 w}{\partial v^2}-2uv\frac{\partial^2 w}{\partial u\partial v}+v^2\frac{\partial^2 w}{\partial u^2}-u\frac{\partial w}{\partial u}-v\frac{\partial w}{\partial v}=0$$ I have to apply the change $w=g(\psi)$ where $g\in \mathcal{C}^2$ and $\psi (u,v)=\arctan\frac{v}{u} $. Only to reduce the equation, I'm not supposed to solve it.

But I have not make a change of this type, usually I make for instance from variables $(u,v)$ to $(x,y)$ but not for the original function $w$. How can I proceed?

Robert Z
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Valent
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1 Answers1

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Just substitute $w$ with $g(\arctan \frac vu)$ in the equation and use chain rule to expand the derivatives. After you get the simplified version, replace $\arctan \frac vu$ with $\psi$

Vasily Mitch
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  • ohh I see, is not a change of variable, instead is just a particular case for a function! – Valent Jul 27 '21 at 14:49
  • The proper way to do it is to change variables of course. But you should be smart what to use as a second variable. Consider, $(u,v)\to(\psi,r)$ where $\psi = \arctan \frac vu$, $r=u^2+v^2$ and use your usual methods – Vasily Mitch Jul 27 '21 at 17:35