Parameterization of the initial condition of nonlinear PDE

Question

(NOTE: I am not asking for the solution to this PDE)

I have the PDE, $$u_x u_y + ln(x^2)=0,$$ with the condition that, $$u(x_0,y)=y,$$ where $x_0$ is a constant. I am to find the explicit solution (using Charpit's equations). In order to do this I first need to parameterize the initial condition in the form of a curve $\Gamma (s)$ in order to solve.

I am used to parameterizing curves but not of this type.

I have written down, $$u=s, x=x_0, y=s,$$ just as a hunch but I have no working to show for it. Is this correct? How do you parameterize equations of this type?

Thanks in advance.

Answer 1

The correct parameterisation is $$u=s, y=s, x=x_0$$ This leads to a solution through Charpit's method of $$u=-2xln(x)+2x+2x_0ln(x_0)-2x_0+y$$