Cauchy problem for partial derivative equation

Question

This my first encounter with PDE. Given the equation $x D_x(z)+(y-xz)D_y(z)=z$, where $z=z(x,y)$ and initial conditions $y-x=2z,zx=-1$. The question is: how to interpret these conditions. I have one guess: $y=x+2z=x-\frac{2}{x}$, this is a curve on $\mathbb R^2$. And on this curve there must be $z=-\frac{1}{x}$. Is it correct interpretation?

Answer 1

Hint:

For the general solution,

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dx}{dt}=x$ , letting $x(0)=1$ , we have $x=e^t$

$\dfrac{dz}{dt}=z$ , letting $z(0)=z_0$ , we have $z=z_0e^t=z_0x$

$\dfrac{dy}{dt}=y-xz=y-z_0e^{2t}$ , we have $y=F(z_0)e^t-z_0e^{2t}=xF\left(\dfrac{z}{x}\right)-xz$ , i.e. $z=xf\left(\dfrac{y}{x}+z\right)$