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I'm trying to solve the following ODE:

$$f'(x)=-f(x)^2+P(x)f(x)^3+Q(x)f(x)^4$$

where the functions $P(x)$ and $Q(x)$ are known.

This nearly resembles Abel Equation of first kind. The power in $f$'s is of 1 order too high. Maybe this equation has a 'canonical form' like in Abel equation of the first kind.

Any hints on solving this? Maybe someone sees a way to reduce it to one of the forms in List of nonlinear ODE

EDIT:

Equivalently, for $f(x)=\frac{1}{g(x)}$:

$$g'(x)=\frac{Q(x)}{g(x)^2}-\frac{P(x)}{g(x)}+1$$

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