We consider the following O.D.E \begin{align}(a)\qquad\qquad\begin{cases}x'(t)=-x(t)+\left[\sin x(t)\right]^2 & t\geq 0,\\x(0)=x_0\in \Bbb{R}&\end{cases}\end{align} I want to prove that $0$ is locally exponentially stable.
MY TRIAL
Let $f(x)=-x+\left[\sin (x)\right]^2.$ Then, $f'(x)=-1+2\sin (x)\cos (x).$ Since $f'(0)=-1<0,$ then $0$ is locally exponentially stable.
Kindly help check if my work is correct. If not, alternative solutions will be highly appreciated. Thanks!