I am trying to solve the following problem:
Problem:
Find conditions on $a, b, c$ such that the solution to the differential equation $$\frac{dy}{dt} = a - by + cyx\;\;,\;\;\;\;\;t>0$$ are bounded. Here $a, b, c$ are positive, real constants,$x, y$ depend on the independent variable $t$, and $x(0), y(0) > 0.$
My attempt:
My initial thought was that the solutions are bounded for all real, positive $a, b, c.$ So I tried to prove that. I tried to show that $\frac{dy}{dt}\leq NiceFunction(t)$ but the term $cyx$ is causing trouble. Can anyone please tell me how to approach this kind of problem and impose conditions on constants so that the solutions are bounded? Any help or advice is highly appreciated! Thank you in advance!!