I have the following differential equation system and I would like to give the p(t), n(t) functions explicitly.
$$\frac{dn}{dt} = (\xi-\alpha)np-\xi n^{2} \quad (1)\\ \frac{dp}{dt} = (\xi-\alpha)np-\xi p^{2} \quad (2)$$
If I express the p(t) variable from the first equation and substitute it to the second equation I get the next differential equation where a, b and c are constants.
$$\ddot{n}n+a\dot{n}n^{2}+b\dot{n}^{2}-cn^{4}=0 \\ a = \frac{\alpha(\xi-\alpha)+2\xi^{2}}{\xi-\alpha}\\ b = \frac{\alpha}{\xi-\alpha}\\ c = \frac{\alpha\xi(\alpha-2\xi)}{\xi-\alpha}$$
If I label the first derivative of n by y then this equation becomes a Chini's ODE:
$$ \dot{n} = y \qquad \ddot{n}=\frac{dy}{dt} = \frac{dy}{dn}\frac{dn}{dt} = \frac{dy}{dn}y\\ ny\frac{dy}{dn}+ayn^{2}+by^{2}-cn^{4}=0\\ \frac{dy}{dn} = -an-\frac{by}{n}+\frac{cn^{3}}{y}$$
In this case the Chini's equation is solvable since the Chini invariant independent from n and we can make the differential eqaution separable with the following transformation.
$$ y(n) = n^{2}v(n)$$
Wolfram alpha gives a complicated solution for v(n) in an implicit equation therefore I can't go on. (I have tried to simplifying it with respect to that the constants a,b and c contains $\alpha$ and $\xi$ and it will be nice but it remains implicit.)
Could someone help me to solve the original differential equation system with an other method? Maybe the system similar to generalized Lotka-Volterra model with two species.