# Method of characteristics

1. Jul 11, 2011

### gtfitzpatrick

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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?

2. Jul 11, 2011

### 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.