Solving First Order ODE using the integrating factor approach

Question

I am trying to solve the differential equation, but I do not understand the method. Here is my working:

Answer 1

From the first diagram-

$\frac{di}{dt} + i = 10t.e^{-t}$

Multiply equation by $e^t$

$\frac{di}{dt}.e^t + i.e^t = 10t$

Now $\frac{di}{dt}.e^t + i.e^t = \frac{d}{dt}(i.e^t)$

So we have $\frac{d}{dt}(i.e^t) = 10t$

Now integrate to solve further.

Step 3-

$\frac{d}{dt}(i.e^t) = i.\frac{d}{dt}(e^t) + e^t\frac{di}{dt}(i)$

= $i.e^t + \frac{di}{dt}.e^t$

Answer 2

The step from line 3 to line 4 isn't correct: $\frac{di}{dt} e^t + ie^t = \frac{d}{dt}(ie^t)$, so line 4 should read $\frac{d}{dt}(ie^t) = 10t$ which can then be integrated directly.

In general, the ideas behind an integrating factor are outlined here as well as I could explain (if not better).