The problem:
Let $A:\Bbb{R} \mapsto M_n (\Bbb{R})$ be continuous and suppose that there exists $L>0$ such that $$ \| A(t)\| \leq L \; ,\; \forall t \in \Bbb{R} \; . $$ And let $x(t)$ be a solution of $x' =A(t) x $
Problem a. Find the equation whose solution is: $$\text{given } \alpha \in \Bbb{R} \; ,\; y_{\alpha} (t)=e^{-\alpha t} x(t)\; . $$
Problem b. Prove the existence of values of $\alpha $ such that $$\lim_{t \to \infty} y_{\alpha} (t) $$ is either zero or infinity.
I have been reading some handouts of Prof. Grant Gustafson, which explain linear systems of differential equations; however, I still don't understand how to find the equation requested in the problem.
I also don't know how can I use the fact that the image is bounded.
How can I find the equation the problem asks for?
Edit: Following the advice on Fred's answer, I fixed the expression involving $\alpha t$ .
You wrote $\alpha \in \Bbb{R}$ and $y_{\alpha} (t)=e^{-\alpha (t)} x(t)$. I think it should read $y_{\alpha} (t)=e^{-\alpha t} x(t)$, since $ \alpha$ is a constant.
Now use the product rule to obtain:
$$y_{\alpha}' (t)=e^{- \alpha t}(A(t)-\alpha I)x(t).$$