Issue solving a linear differential equation.

Question

I am currently working through a chapter in my textbook, and it is on solving a form of linear differential equation. I have been trying to figure out what is happening for about 3 hours as of now.

I don't understand how $\frac{d}{dt}e^{k(t))}x = 0$, since it says that $\frac{d}{dt}e^{k(t)} = p(t)e^{k(t)}$. Surely this would mean that$\frac{d}{dt}e^{k(t)}x = x*p(t)e^{k(t)}$.

Answer 1

Start from $e^{k(t)}\frac{dx}{dt}+p(t)e^{k(t)}x=0$

Note that $p(t)e^{k(t)}x=\frac{d}{dt}e^{k(t)}$

Therefore, by the product rule

$\frac{d}{dt}(e^{k(t)}x)=\frac{d}{dt}(e^{k(t)})*x+\frac{dx}{dt}e^{k(t)} = p(t)e^{k(t)}x+e^{k(t)}\frac{dx}{dt}=0$

Let me know if this clears up the confusion

Answer 2

Note that \begin{align*} \frac{d}{dt}\left(e^{k(t)}x\right) &= e^{k(t)}\frac{dx}{dt} + \frac{d}{dt}\left(e^{k(t)}\right)x\\ &= e^{k(t)}\frac{dx}{dt} + p(t)e^{k(t)}x \end{align*}

according to the product rule. So if we run through the lines in reverse order, we have the derivation.