Question

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(a) Find the Laplace transform of the solution of the initial-value problem y" - 4y + 3y = -3x + 2 cos(3x), y(0) = 2, y (0) = 3. 8² +68 is the Laplace transform of the solution of an intitial-value problem. Find the (8 + 1)(82 +9) solution y = y(a) by finding the inverse transform of Y.

Answer 1

<b> Y C8) = 82768_ C8+1) (5249) Using pooktad faraeldon $2768 (8+1)(82+3) 2C8+1.9882+9) com tela = -1 + 38+9 i į L 8+1. 8279 847 Now I'{ycv8 = -["{*} +31 () olw + ["{ycs)} = et + 3 Col3 + + x sling! Which is required inverse transform of Y ma دنيا ตาม fince I'expe} = logat I'lgtares = dinah I'foto} = eat

" Find the Laplace torane form of the sals of the IV p. Y" - 44'+3y = -3x +2 C08 3X , Y607 = 2 Taking Labulaice foraine form of beth side. Y'(0)=3 [ (7") – 4 L(') +3 L(Y) = -31(x) +2 [ {los3x} 82568) – 84(0)-4'0) - 4$ 8FCB) – f(0)} +35(B) = -3 +2 lights (ge-48 +3) F(R) – 8 x2 3 + 462) = 3 + 4 (82_48 +3) F68) – 28 – 3 +8 = -2+284949 (824843)F(8) = +28 -5 = 283 – 382 27 #28-5) (844982) | (879). 82 = 283 -382-27 +. 285_584 +1882-4584 82(82+9). (8248+3) ? c&_ 285-584 +2083-4882-27 o 82 (82+9) 768) 285-584+2083-4889-27 (8-1) (8-3) 84(?+9) Here I $ f(x)} = f c8) f(x) = ["{fC8 ).

Taking In verse Laplace Transform we get f(x) = ['SÉ (8)} Here We Use. [{fim} = f(B) & Then [$f'xi= 8 $(8) – f(0) [$€"(x)} = $?(8) – 8€ (0) - f'(0) IF