I have a question about the proof of this elementary DE, the calculations are easy
$$\frac{\text{d}y}{\text{d}t}+a(t)y=0\rightarrow \left | y(t)\right|=\text{exp}(-\int a(t)\text{d}t+ca)\rightarrow |c\,\text{exp}(-\int a(t)\text{d}t)|=c$$
-not sure if you can multiply here by the integral and subsume it into the absolute value.
$y(t)exp(\int a(t)\text{d}t)$ is a continuous function of time
-I'm not sure how this is assumed to be continuous in time?
If the absolute value of a function $g(t)$ is constant then $g$ must be constant,
Proof:
1.If $g$ isn't constant there exists $t_1,t_2$ for which $g(t_1)=c$, $g(t_2)=-c$.
-if g(t)=x or d then why does g necessarily need to equal c or -c?
2.By the intermediate value theorem g must achieve all values between $-c$ and $c$ which is impossible if $|g(t)|=c$