Solution of the the following

7. LAPLACE TRANSFORMS OF DERIVATIVES THEOREM. The Laplace transform of the derivative f'(t) exists when s > a, and L(f') = SL(f) - f(0) L(fn) = s"c(f) - sn-1f(0) - sn-2f' (0) - ... - fn-1(0) SAMPLE PROBLEM: Find the Laplace transform of f(t) = sint using the transform of derivatives. Solution: f = sin't ... ... ... ... ... .. f (0) = 0 f' = 2 sin t cost f' = sin 2t ... ... ... ... .....f (0) =0 f" = 2 cos 2t Hence, [(f") = s2 [( f) - sf (0) - f'(0) L(2 cos 2t) = s'L(f) - s(0) - 0 2L(cos 2t) = s2 L(sin't) 2 S 152 + ( 2)2 = 52 [(sin2t ) Since, L(cos 2t) =- $2+4 2s L(sin t) = - $2 ($2 + 4) Therefore, 2 L(sin't) = S ($2 + 4)E. LAPLACE TRANSFORMS OF DERIVATIVES Find the Laplace transforms of the following functions using the transform of derivatives. 1) f(t) = t Ans. $2 2) f(t) = cos22t Ans. $2 + 8 s($2 + 16) 3) f(t) = te-71 Ans. (s + 7) 2 4) f(t) = e2t cos 5t Ans. s - 2 (s - 2)2 + 25 5) f(t) = (sin t - cos t)2 Ans. $2 - 2s + 4 s(s2 + 4)