PDE i.v.p. using method of characteristics

  • Thread starter sunrah
  • Start date
  • #1
199
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Homework Statement


solve x2ux + y2uy = 0 for u(2,y) = y

Homework Equations


The Attempt at a Solution



with a = x2 and b = y2

y' = b/a = (y/x)2 this can be solved for y by separation of variables:

[itex]y = \frac{x}{1-xC}[/itex]
and
[itex]C = \frac{y-x}{xy}[/itex]

now

[itex]u(x,y) = f(C) = f(\frac{y-x}{xy})[/itex]

applying initial value conditions

[itex]u(2,y) = f(\frac{y-2}{2y})[/itex]

this is where my understanding runs out. how do i determine f according to the initial value? i have looked at several books but they just assume this step is obvious
 
Last edited:

Answers and Replies

  • #2
199
22
have solved it now :smile: by taking a different approach and by converting to a system of ODEs and doing a coordinate transform from x(t) -> x(t,s) but I would still like to know how to solve it according to my original question, thanks
 

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