First Order ODE/x y' = Root of (x^2 + y^2)

Theorem
The first order ordinary differential equation:


 * $(1): \quad x y' = \sqrt {x^2 + y^2}$

is a homogeneous differential equation with solution:


 * $3 x^2 \ln x = y \sqrt {x^2 + y^2} + x^2 \map \ln {y + \sqrt {x^2 + y^2} } + y^2 + C x^2$

Proof
We divide through by $x$ to show that $(1)$ is homogeneous:

By Solution to Homogeneous Differential Equation, its solution is:
 * $\ds \ln x = \int \frac {\d z} {\map f {1, z} - z} + C$

where:
 * $\map f {x, y} = \dfrac {\sqrt {x^2 + y^2} } x$

Thus: