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I have the following differential equation that I try to find a general solution :

$$\frac{dx\left(t\right)}{dt}=x\left(t\right)\left(1-x\left(t\right)\right)-ay\left(t\right)$$

Since there is the variable $x^2$, I tried to transform the variable in order to have a simpler one but it did not work. How can I solve this equation of where should I look in order to have an intuition to solve it ?

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    $\begingroup$ This is a special case of the Riccati equation. $\endgroup$ – projectilemotion Oct 8 '17 at 16:14
  • $\begingroup$ This DE is called the logistic equation with harvest and there is no explicit solution for general $y(t)$. $\endgroup$ – xpaul Oct 8 '17 at 16:16
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    $\begingroup$ They are two unknown functions $x(t)$ and $y(t)$ in only one equation. You expect solve the equation for what ? $\endgroup$ – JJacquelin Oct 8 '17 at 16:56
  • $\begingroup$ @JJacquelin I try to solve it for $x(t)$ $\endgroup$ – optimal control Oct 9 '17 at 14:47
  • $\begingroup$ OK. So, $y(t)$ is a known function and the equation is a Riccati ODE as already mentioned in the comment from projectilmotion. See Eq.(4) in mathworld.wolfram.com/RiccatiDifferentialEquation.html and follow the method to transform into a linear ODE. $\endgroup$ – JJacquelin Oct 9 '17 at 19:10

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