Question

Question #6cafa

Answer 1

`y'' - y' - 2y =-12(x + 2)`

Explanation:

Given: `y = e^(2x) + 3(2x+3) = e^(2x) +6x + 9`

Find the first derivative using `(e^u)' = u' e^u`:

Let `u = 2x; " " u' = 2`

`y ' = 2e^(2x) + 6`

`y'' = 4 e^(2x)`

Find `" "y'' - y' - 2y`:

`y'' - y' - 2y = 4 e^(2x) - ( 2e^(2x) + 6) - 2(e^(2x) +6x + 9)`

Distribute the negatives:

`y'' - y' - 2y = 4 e^(2x) - 2e^(2x) - 6 -2e^(2x) - 12x -18`

Simplify by combining like-terms:

`y'' - y' - 2y = -6 - 12x - 18 = -24 -12x`

Factor: `" "y'' - y' - 2y =-12(x + 2)`