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Let $f$ be a continuous function.

Can anyone please help me out to find the solution of the ODE: $$y\left( \frac{dy}{dx} + a y + b\right) = f(x)$$

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$$\frac{dy}{dx} + a y + b = \frac{f(x)}{y}$$ Let $\quad y(x)=\frac{1}{u(x)}\quad\to\quad -\frac{u'}{u^2}+\frac{a}{u}+b=u(x)f(x)$ $$u'=a\,u+b\,u^2-f(x)u^3$$ This is an Abel's differential equation of the first kind. https://en.wikipedia.org/wiki/Abel_equation_of_the_first_kind

The solvability depends on the kind of function $f(x)$. https://www.hindawi.com/journals/ijmms/2011/387429/#sec2

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There doesn't seem to be a closed-form solution in general, or even in some simple special cases such as $f(x)=x$.

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