Proving solution to ODE is decreasing

Question

If I have an equation of the form $y’=A(x)y+B(x)$ with an initial condition $y(0)=0$ where $A(x),B(x)$ are negative and decreasing for all $x>0$ with the following limits being finite and negative: $$\lim_{t\rightarrow \infty} A(x),B(x)=L,M .$$, how can I show that the function solution $y(x)$ is decreasing? The functions $A(x),B(x)$ I cannot express them in an explicit way unfortunately.

So my attempt so far is: Solving the equation generally yields easily that $y(x)$ is a negative solution. I also know that $e^{\int A(x) dx}y(x)$ is decreasing too. Because $B(x)$ is negative then for some “initial” interval $[0,x_0]$ I have that $y’(x)<0%$ so I was thinking to prove by contradiction that there’s an interval $[x_0,x_1]$ where $y’(x)>0$ by comparing maybe to the equation $y’=A(x)y(x)$ but not sure how to approach.

Help is appreciated

Answer 1

I don't think that is true? Consider the ODE $y'(x)=-e^xy(x)-x$ with solution $$y(x)=e^{-e^x}\int_0^x-e^{e^t}t\mathrm{d}t$$ where $$y'(x)=e^{x-e^x}\int_0^xe^{e^t}t\mathrm{d}t-x$$ computing numerically, is positive at $x=2$. And finite limits can be achieved by connecting $-e^x$ and $x$ with functions that do converge at infinity.