What is the solution for a PDE using method of characteristics?

[gtfitzpatrick]

Homework Statement

solve using method of characteristics
$$<br /> y \frac{ \partial u}{ \partial x} - x \frac{ \partial u}{ \partial y}= 1<br />$$ where u(x,0) = 0 for 0<x<infinity

The Attempt at a Solution

$$<br /> \frac{ \partial y}{ \partial x} = \frac{x}{y}<br />$$ which gives $$y^2+x^2= k$$ the projected characteristic. but its the second part that is giving me trouble when i go to find $$<br /> \frac{ \partial u}{ \partial x} = \frac{1}{y}<br />$$ which doesn't work out right...any suggestion anyone?

 

[tiny-tim]

hi gtfitzpatrick!

(have a curly d: ∂ )

∂y/∂x = x/y gives y² minus x² = k

 

[gtfitzpatrick]

hi gtfitzpatrick!

(have a curly d: ∂ )

∂y/∂x = x/y gives y² minus x² = k

my mistake it should have read

∂y/∂x = -x/y gives y² + x² = k

but don't know how to work the second part?

 

[gtfitzpatrick]

oh and hi back