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

Theorem
The first order ODE:
 * $(1): \quad 2 x y \, \mathrm d x + \left({x^2 + 2 y}\right) \, \mathrm d y = 0$

has the solution:
 * $y \left({x^2 + y}\right) = C$

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

where:
 * $P \left({y}\right) = \dfrac 1 {2 y}$
 * $Q \left({y}\right) = -1$
 * $n = -1$

and so is an example of Bernoulli's equation.

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

where:
 * $\mu \left({y}\right) = e^{\left({1 - n}\right) \int P \left({y}\right) \, \mathrm d y}$

Thus $\mu \left({x}\right)$ is evaluated:

and so substituting into $(3)$: