# Solving Non-Linear First order ODEs

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.

• 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? – user653734 Mar 13 at 11:03

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 $$\alpha,\beta,\delta,\kappa,\mu,\nu$$.