solve: Uxy = -Ux using indicated transformation v=x and z=x+y

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- May 15th 2014, 03:02 AM #1

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- May 15th 2014, 04:02 AM #2

- May 15th 2014, 04:31 AM #3

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## Re: PDE please help!!

I agree with Prove It. The equation, as given, can be directly integrated while making the "indicated transformation" gives a much harder equation!

But if you must transform, use the fact that, for any function, F(x,y), $\displaystyle \frac{\partial F}{\partial x}= \frac{\partial F}{\partial v}\frac{\partial v}{\partial x}+ \frac{\partial F}{\partial z}\frac{\partial z}{\partial x}= \frac{\partial F}{\partial v}+ \frac{\partial F}{\partial z}$ and $\displaystyle \frac{\partial F}{\partial y}= \frac{\partial F}{\partial v}\frac{\partial v}{\partial y}+ \frac{\partial F}{\partial z}\frac{\partial z}{\partial y}= \frac{\partial F}{\partial z}$

So, letting $\displaystyle F= U$, $\displaystyle U_x= U_v+ U_z$ and, letting $\displaystyle F= U_v+ U_z$, $\displaystyle U_{xy}= U_{vz}+ U_{zz}$.

The equation $\displaystyle U_{xy}= -U_x$ becomes $\displaystyle U_{vz}+ U_{zz}= -U_v- U_z$.

As Prove It said, the original equation is much easier to solve than the transformed equation!