Bernoulli's Equation/2 x y dx + (x^2 + 2 y) dy = 0

Theorem
The first order ODE:
 * $(1): \quad 2 x y \rd x + \paren {x^2 + 2 y} \rd y = 0$

has the solution:
 * $y \paren {x^2 + y} = C$

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

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 $\map \mu x$ is evaluated:

and so substituting into $(3)$: