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I want to solve the following Bernoulli differential equation: $$A'(t)=A^2(t)[-2\sigma +1]-2aA(t)$$ where $\sigma$ and $a$ are real numbers.

Until now I have divided both sides of the equation with $A^{2}(t)$ and defined $u=A^{-1}(t)$ and $u'=-A^{-2}(t)A'(t)$. The new equation that arises is $$u'+2au=1-2\sigma $$

It must be relatively simple but how do I solve this? Can someone give me some hints? Also can someone propose some good books about stochastic control and advanced algebra (including Ricatti equations and differential equations etc)?

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    $\begingroup$ Best solved using the integrating factor method- multiply both sides by $e^{2at}$, and the LHS is then just the differential of u times the integrating factor. $\endgroup$ Commented Jul 10, 2021 at 22:37
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    $\begingroup$ Thank you very much! It was really simple after all...I was stuck for hours. $\endgroup$ Commented Jul 10, 2021 at 22:45

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