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5. Find f(t) if f(t) 2 [* f(x) dx = 2t. Answer: - Let L{f(t)} = F(s). Take the Laplace transform of both parts of the equation: - F(s) 2 - F(s) = 2 1 $2 F(s) (1-2) = 2 = F(s) = To - S 2 s(s-2) invert F(s), one can use (a) partial fractions or (b) {f(x) dr} = F(s) (a) F(s) 2 = s(s - 2) S 2 1 (b) s(s2) S B + 1 S 2 y(t) = 1 + e2t. F(s). Then L{F(s)} = L = 2 S { 2} = 2e2t = f(t), and S t y(t) = f * f (x)dx = 2 * t S e2x dx = 2. 1 e2x 2. e | 6 = e - 1. 2 66