1. x'' - 2x' + 2x = 0 ; x(0) = 0 ; x'(0) = 1 L[x''] - 2L[x'] + 2L[x] = L[0] s^2 X(s) - sx(0) - x'(0) - 2(sX(s) - x(0)) + 2X(s) = 0 s^2 X(s) - sx(0) - x'(0) - 2sX(s) + 2x(0) + 2X(s) = 0 s^2 X(s) - 2sX(s) + 2X(s) - 1 = 0 X(s) = 1 / (s^2 - 2s + 2) Complete the square: s^2 - 2s + 2 : (s - 1)^2 + 1 L^-1 [1 / ((s - 1)^2 + 1)] ; k^2 = 1 k = ? = 1 ? e^t sin t The answer is x(t) = e^t sin t. 2. x'' + 0.4x' + 2x = 1 - h_5(t) ; x(0) = 0 ; x'(0) = 0 L[x''] + 0.4L[x'] + 2L[x] = L[1 - h_5(t)] s^2 X(s) - sx(0) - x'(0) + 0.4(sX(s) - x(0)) + 2X(s) = L[1 - h_5(t)] s^2 X(s) + 0.4sX(s) + 2X(s) = 1/s - 1/s e^-5s
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