Bernoulli's Equation/x y' + y = x^4 y^3

Theorem
The first order ODE:
 * $(1): \quad x y' + y = x^4 y^3$

has the general solution:
 * $\dfrac 1 {y^2} = - x^4 + C x^2$

Proof
Let $(1)$ be rearranged as:
 * $(2): \quad \dfrac {\d y} {\d x} + \dfrac 1 x y = x^3 y^3$

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

where:
 * $\map P x = \dfrac 1 x$
 * $\map Q x = x^3$
 * $n = 3$

and so is an example of Bernoulli's equation.

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

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

Thus $\map \mu x$ is evaluated:

and so substituting into $(3)$:

Hence the general solution to $(1)$ is:


 * $\dfrac 1 {y^2} = - x^4 + C x^2$