Question

I am unable to get the proper inverse for the last one. Any help is appreciated!

Entered Answer Preview Result (s2 +36) Y() -8s-7 comect comect s2 +36 8s [s/( (gh 2)+36י2[+[)8.gl|(s*2)+361l+(7Ms^2)+36]} comect (s2+36) 36 +36 8 cos(7t) 7 sin(7) +sin(7t) 8 cos7t)+7 sin(7t+(t/12) sin(7t) incorrect 12 At least one of the answers above is NOT correct. (1 point) Consider the initial value problem y" + 36y-cos(6t), y(0)-8, y,(0)s7 a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). (s 2+36)Y(s)-8s-7 help (formulas) b. Solve your equation for Y(s). c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t).

Answer 1

L { sin 6 t } = 6/s _2 + 36 = > L { t sin 6 t } = (-1) ^1 d/ds (6/s^2 + 36 _ = (6) 2 s/(s^2 + 36)^2 = 12 s/(s^2 + 36)^2 therefore L^-1 { s /(s^2 + 36)^2 = t sin 6 t/12 L^-1 { 8 s/ s^2 + 36 } = 8 cos 6 t L^-1 { 7/s^2 + 36 } = 7/6 sin 6 t therefore L^-1 { s/(s^2 + 36)^2 + 8 s/s^2 + 36 + 7/s^2 + 36 = y (t) = t sin 6 t/12 + 8 cos 6 t + 7 /6 sin 6 t