We are given the following system of differential equations
$$\begin{aligned} x' &= a_1xy\\ y' &= a_2 xy-by\end{aligned}$$
where $a$ and $b$ are constants(let us assume them positive).
I know how to find its equilibrium/fixed points and checking stability. But what I want to know is whether there is any method that will yield $x(t)$ and $y(t)$.
If so, then please let me know about it. I am not aware about this thing. Just like there is a fixed procedure to solve linear differential equations of the form $y'+Py=Q$. Is there any concept or method of formula to solve the system of differential equations above.
Answer:
$$ (1) : x'=axy $$
$$(2): y'=axy-by $$
Then $$(1)-(2) : y'+by=x'$$ and $$(1) : x'/x=ay =>(Lnx)/a= ∫ydt $$ So $$∫((1)-(2))dt$$ $$y=x-b(Lnx)/a+C$$ $$(1): x'=ax^2-bx(Lnx)+C'x => dx/dt=ax^2-bx(Lnx)+C'x $$ $$y(t)=x(t)-b(Lnx(t))/a+C$$
C and C' are constant.
Note: if there any mistakes in the answer, feel free to edit in the corrections or comment them.