Question

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. y'' − 4y' + 13y = 0; y = e2x cos 3x When y = e2x cos 3x, y=?

Answer 1

Answer:

y = e^(2x) cos 3x is a explicit solution of y'' − 4y' + 13y = 0

Step-by-step explanation:

y'' − 4y' + 13y = 0   (1)

Verify that y = e^(2x) cos 3x is a explicit solution. We derive this solution:

y'=2e^(2x) cos 3x-3e^(2x) sin 3x     (2)

y''=4e^(2x) cos 3x - 6e^(2x) sin 3x -6e^(2x) sin 3x -9e^(2x) cos 3x

y''=-5e^(2x) cos 3x - 12e^(2x) sin 3x   (3)

In order to verify, we replace (2) (3) in (1):

$-5e^{2x} cos 3x - 12e^{2x} sin 3x -4[2e^{2x} cos 3x-3e^{2x} sin 3x ]+13{e^}2x} cos 3x=(-5-8+13)e^{2x} cos 3x +(-12+12)e^{2x} sin 3x=0$