Consider the initial value problem y' + 2y =12t, y(0) = 9. 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). help (formulas) b. Solve your equation for Y(s). Y(s) = Lly(t)} c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). y(t) =solution - Given : y- 1- 24 = 127 , ylo ) =g (" ) Now Applying laplace framsform to equation ( 1) on both side we get - L [ . y' + 24 ] = L [ 12- ] sy (s ) - y( o) + 2 4 ( s ) = 12 Answer 82 poult (a) (b) s y (8 ) + 2 718 - 9= 12 8 2 ( 272 ) Y( 8 ) = 12 -1. 9 82 12 84 2 By applying partial traction we get 6 3 's -f- 6 2 .1- 2 & -.. . YUs = G 12 $ 2 -2 8 -. 2 answer of(C) Now we will tranform apply Inverse laplace on Y's = -2 3 + 8 to find our solution y($ ) L ' [ Y 18) ] = LY[ 6 12 $ 2 + 8+ 2 8 + 2 y(+) = 67 - 3+ 12 p-27 Hence , solution of given differential equation is - 20 Y () = Ize + 67 - 3 This I's answer of port (c )