In the book on p.68 is a system of differential equations for a Predator-Prey model (Lotka-Volterra) given as:
$$ \dot x=x(\alpha-c\gamma) \\ \dot y=y(\gamma x -\delta) $$
On the next page, it is said that the system has solution curves (the differential equation is separable) given by $x^\delta y^\alpha e^{-\gamma x} e^{-cy} = K$, where $K$ is a constant.
I tried to do the separation of variables like this:
$$ \frac{dy}{dx}=\frac{y(\gamma x -\delta)}{x(\alpha-c\gamma)} \\ \frac xy = \frac{dx(\gamma x -\delta)}{dy(\alpha-c\gamma)} \\ $$
But the integration is not easy at all
$$\int \frac {dy(\alpha-c\gamma)}y = \int \frac{dx(\gamma x -\delta)}x$$
So I tried using MATLAB as dsolve([diff(x)==x*(a - c*y), diff(y)==-y*(d - g*x)]) which resulted to Warning: Explicit solution could not be found.
Can anyone please tell me how to get that solution in bold by hand, by MATLAB or both ways please?
Separation of variables is fine.
Note $$\begin{align} \dfrac{dy}{dx} = \dfrac{y( \gamma x-\delta)}{x(\alpha - cy)} &\implies \displaystyle\int^y\left(\dfrac{\alpha}{y'} - c \right)dy' = \displaystyle\int^x \left(\gamma -\dfrac{\delta}{x'}\right)dx' \\ &\implies \alpha\log y-cy=\gamma x-\delta \log x+\log K \\ &\implies \exp[\log x^\delta y^\alpha -\gamma x - cy] = K\\ &\implies \boxed{x^\delta y^\gamma e^{-\gamma x}e^{-cy} = K} \end{align}$$