# Method of Characteristics help

1. Sep 22, 2011

### lackrange

Problem: Find the characteristics of
$$xyu_x+(2y^2-x^6)u_y=0$$
So I rewrote this as $$u_x+\frac{2y^2-x^6}{xy}u_y=0$$ and then set this as $$\frac{du}{dx}=0\implies \frac{dy}{dx}=\frac{2y^2-x^6}{xy}$$
I solved this, and found that the characteristics were $$\frac{y^2+x^6}{x^4}=C$$
where C is a constant, and u is constant along this curve. Now the problem says consider the initial condition $$u(x,α x^n)=x^2,\;\;n\in \mathbb{N}\;\;α>0,$$
for what α>0 does the problem have a solution? For what α > 0 is the solution uniquely? Your answer may depend on n (Try n=1, n=2 etc.).

So I wrote $$αx^n=\frac{y^2+x^6}{x^4}$$ and solved for α, but I don't think this is what I am suppose to do, can someone help me please?

2. Sep 27, 2011

anyone?