Method of Characteristics for Solving Non-Divergent Differential Equations

[gtfitzpatrick]

Homework Statement

x($\partial u / \partial x$) + y($\partial u / \partial y$) = -$x^2u^3$where 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.