There was an algebraic error in the lecture. Here is a correction.


The problem was to solve  y' = -y + exp(-2t)  + 1.


1) We solve  y' = - y + exp (-2t)   by substituting in   y = A(exp(-2t)
and solving for A.  After the first step, we have:

        y' = -2A(exp(-2t)= -y + exp(-2t)=  -Aexp(-2t)  + exp (-2t).

Cancel out the exp(-2t) and manipulate algebraically to get.

            -A =  D-1  {Not -3A = 1 as claimed in lecture).

The general solution for this equation is  y = k exp(-t) - exp(-2t).

2)  The general soution to  y' = -y + 1  is  y = k exp(-t) +1.

3) By the superposition principle for linear equations, the general
solution of the original equation is

        y' = k exp(-t) - exp(-2t) + 1.