Riccati Equation/y' = (y over x) + x^3 y^2 - x^5

Theorem
The Riccati equation:
 * $(1): \quad y' = \dfrac y x + x^3 y^2 - x^5$

has the general solution:
 * $C \exp \dfrac {2 x^5} 5 = \dfrac {y - x} {y + x}$

Proof
It can be seen by inspection that:
 * $y_1 \left({x}\right) = x$

is a particular solution to $(1)$.

Thus from General Solution of Riccati Equation from Particular Solution:
 * $y = x + z \left({x}\right)$

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

From Bernoulli's Equation: $y' - \left({\dfrac 1 x + 2 x^4}\right) y = x^3 y^2$:


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

Thus: