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:


 * $3 x^2 \ln x = y \sqrt{x^2 + y^2} + x^2 \ln \left({y + \sqrt{x^2 + y^2} }\right) + y^2 + C x^2$

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

By Solution to Homogeneous Differential Equation, its solution is:
 * $\displaystyle \ln x = \int \frac {\mathrm d z} {f \left({1, z}\right) - z} + C$

where:
 * $f \left({x, y}\right) = \dfrac {\sqrt{x^2 + y^2} } x$

Thus: