<s>[INST] Inequality with cosine and sine Let $A:=f^2+g^2$, where $f,g$ are functions of $x$ such that
$$f'=(c-1)(f\cos(x)\sin(x)+g\sin^2(x)),$$
$$g'=-(c-1)(f\cos^2(x)+g\cos(x)\sin(x)),$$
for some constant $c$.
(Note: $f'\cos(x)+g'\sin(x)=0$)

How do I show that $A'\leq4|c-1|A?$

I see that
\begin{align}
A'&=2ff'+2gg'\\
&=2(c-1)\left(f^2\cos(x)\sin(x)+fg\sin^2(x)-fg\cos^2(x)-g^2\cos(x)\sin(x)\right)\\
&=2(c-1)(f\cos(x)+g\sin(x))(f\sin(x)-g\cos(x)).
\end{align}
But how does the desired inequality follow?
 [/INST] Cauchy-Schwartz' inequality tells you that
$$
|a f + b g| \le \sqrt{a^2 + b^2}\sqrt{f^2+ g^2}
$$
Starting from your result, we obtain
$$
A^\prime\le |A^\prime|\le 2 |c-1|1.\sqrt{f^2+g^2}1.\sqrt{f^2+g^2} = 2 |c-1|A
$$
</s>