PDE by method of characteristics

Question

How do I solve this PDE by method of characteristics?

$$2y{\partial z \over \partial x} - {\partial z \over \partial y} = x$$

Answer 1

$$2y{\partial z \over \partial x} - {\partial z \over \partial y} = x$$ Solution written as concisely as the question is : $$\frac{dx}{2y}=\frac{dy}{-1}=\frac{dz}{x}$$ $\frac{dx}{2y}=\frac{dy}{-1}\quad\to\quad x+y^2=c_1$

$\frac{dy}{-1}=\frac{dz}{x}=\frac{dz}{c_1-y^2}\quad\to\quad z+c_1y-\frac 13 y^3=c_2\quad\to\quad z+xy+\frac 23 y^3=c_2$ $$z=-xy-\frac 23 y^3+F(x+y^2)\quad\text{any function }F$$