Suppose we have a first order linear PDE of the form:(adsbygoogle = window.adsbygoogle || []).push({});

a(x,y) u_{x}+ b(x,y) u_{y}= 0

Thendy/dx = b(x,y) / a(x,y)[assumption: a(x,y) is not zero]

The characteristic equation for the PDE is

b(x,y) dx - a(x,y) dy=0

d[F(x,y)]=0

"F(x,y)=constant" are characteristic curves

Therefore, the general solution to the PDE is u(x,y)=f[F(x,y)] where f is an arbitrary function.

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I don't understand the parts in red.

1) Why would dy/dx = b(x,y) / a(x,y) ? This doesn't seem obvious to me at all...how can we derive (or prove) it?

2) Also, what is the meaning of the equation d[F(x,y)]=0?

Thanks for explaining!

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# First order linear PDE

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