- #1
kingwinner
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Suppose we have a first order linear PDE of the form:
a(x,y) ux + b(x,y) uy = 0
Then dy/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!
a(x,y) ux + b(x,y) uy = 0
Then dy/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.
===========================================
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!