First Order ODE/x y dy = x^2 dy + y^2 dx/Proof 1

Proof
Let $(1)$ be rearranged as:
 * $(2): \quad \dfrac {\d x} {\d y} - \dfrac x y = -\dfrac {x^2} {y^2}$

It can be seen that $(2)$ is in the form:
 * $\dfrac {\d x} {\d y} + \map P y x = \map Q y x^n$

where:
 * $\map P y = -\dfrac 1 y$
 * $\map Q y = -\dfrac 1 {y^2}$
 * $n = 2$

and so is an example of Bernoulli's equation.

By Solution to Bernoulli's Equation it has the general solution:
 * $(3): \quad \displaystyle \frac {\map \mu y} {x^{n - 1} } = \paren {1 - n} \int \map Q y \, \map \mu y \rd y + C$

where:
 * $\map \mu y = e^{\paren {1 - n} \int \map P y \rd y}$

Thus $\map \mu y$ is evaluated:

and so substituting into $(3)$: