Method of Characteristics for Solving Non-Divergent Differential Equations

[gtfitzpatrick]

Homework Statement

x(\partial u / \partial x) + y(\partial u / \partial y) = -x^2u^3where u(x,1) = x for -\infty < x < \infty

Homework Equations

The Attempt at a Solution

dy/dx = y/x

= ln(y)=ln(x)+k k=constant of integration
=y = x + e^K
=y=x+k

along this characteristic
du/dx = -(x^2u^3)/x

= -xu^3

= 1/(2u^2) = ln(x) + F(K)

not sure where to go from here...

should i simplify more for u and swap in k=y-x then use the conditions?

 

[Herr Malus]

You're using the wrong method here, it seems. The method of characteristics you've set up is tailored to the case where the divergence of the function u is zero. Here it is not. Try something like a change of variables, to eliminate (say) y from your equation and reduce it to one variable.