Bernoulli's Equation/y' - (1 over x + 2 x^4) y = x^3 y^2

Theorem
The first order ODE:
 * $(1): \quad y' - \left({\dfrac 1 x + 2 x^4}\right) y = x^3 y^2$

has the solution:
 * $y = \dfrac {2 x} {C \exp \left({-\dfrac {2 x^5} 5}\right) - 1}$

Proof
It can be seen that $(1)$ is in the form:
 * $\dfrac {\mathrm d y}{\mathrm d x} + P \left({x}\right) y = Q \left({x}\right) y^n$

where:
 * $P \left({x}\right) = -\left({\dfrac 1 x + 2 x^4}\right)$
 * $Q \left({x}\right) = x^3$
 * $n = 2$

and so is an example of Bernoulli's equation.

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

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

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

and so substituting into $(3)$:

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


 * $y = \dfrac {2 x} {C \exp \left({-\dfrac {2 x^5} 5}\right) - 1}$