Consider the system
$x' = y − x^3$
$y' = -0.5\,x − y + d\sin(t)$
where $d$ is a constant. Show that all solutions exist on $[0,\infty)$. I'm trying to show that any solution is bounded, by showing that the function $x(t)^2 + y(t)^2$ is decreasing. But I end up with $4\,d\,y$.
Use $V=\frac12x^2+y^2$ as Lyapunov function. Then \begin{align} \dot V f&=x(y-x^3)+2y(-0.5x-y+d\sin(t))=-x^4-2y^2+2dy\sin(t) \\ &\le -x^4-y^2+d^2=-V-\left(x^2-\frac14\right)^2+d^2+\frac14 \end{align} using $2dy\sin(t)\le y^2+d^2\sin^2(t)\le y^2+d^2$.
Now consider the region $V>1+d^2$ to get that the vector field is inward pointing. Or use directly that along trajectories in that direction $$ V(x(t))\le e^{-(t-t_0)}V(x(t_0))+\left(d^2+\frac14\right)\left(1-e^{-(t-t_0)}\right). $$