Substitute, simplify, and then solve for Z

[Addez123]

Homework Statement

Simplify the equation
$$x^2 z_x - z_y = 0, x > 0$$
using the variable change u and v.
Then solve the differential equation.

Relevant Equations

$$x^2 z_x - z_y = 0, x > 0$$
$$u = x$$
$$v = 1/x - y$$

By substitution I've solved that $$z_u = 0$$ (which is correct according to textbook)
which leads me to try find z by integrating
$$z = \int z_v \,(dv ?)$$
and I'm just stuck here. Dont know how to integrate that.

 

[Orodruin]

It is not $z_v$ you should integrate (you do not have an expression for it), it is $z_u$.

 

[Addez123]

So $$z = \int z_u \,du = \int 0 \,du = g(v)$$
Doing this and I havn't really solved anything.

 

[Orodruin]

Yes you have. Any (differentiable) function on the form $g(v) = g(1/x - y) = z(x,y)$ solves the differential equation. You can never fully specify the function without boundary conditions. For example, in this case, a possible boundary condition could be $z(x,0) = h(x)$, where $h$ is some known function, which would identify $g(1/x) = h(x)$ and therefore $z(x,y) = h(1/(1/x-y))$.

 

[Addez123]

It did turnout that g(v) was the correct solution :P
Thanks!