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The equation is: $$\frac{dx}{dt} = \beta + {\alpha}x{(1 - \frac{x}{\kappa})} - x(\mu + \nu + \delta)$$

$\beta, \alpha, \kappa, \mu, \nu, \delta$ are all constants. I am trying to solve this differential equation.

I understand that this is a non-linear, first order differential equation, however it is non-separable due to the logistic component. I don't think it is exact either so I don't know what to do now.

Any help would be appreciated.

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  • $\begingroup$ Am I right to say that it takes the form of a Riccati differential equation if the logistic component is expanded out? Is there a technique used to solve Riccati differential equations? $\endgroup$ – user653734 Mar 13 at 11:03
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The equation is nonlinear, but it is separable, as it has the form $$ \frac{\text{d} x}{\text{d} t} = f(x). $$ Therefore, all you have to do is to solve the equation $$ \int^x \frac{1}{f(\hat{x})}\text{d}\hat{x} = t + t_0 $$ for $x$. You will obtain something of the form $$ x(t) = c_1 + c_2 \,\text{tanh} \left(c_3(t+t_0)\right), $$ where $c_1,c_2,c_3$ are constants depending on the model parameters $\alpha,\beta,\delta,\kappa,\mu,\nu$.

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