Solve subject to x(0) = 0, x\'(0) = 0, y(0) = 0, y\'(0) = 0 using Laplace transf

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Explanation / Answer

Apply L .T. ........ s^2X[s]-sx(0)-x'(0)+s^2Y[s]-sy(0)-y'(0) = 1/(s-2) ==> s^2X[s] + s^2Y[s] = 1/(s-2) ---->(1) similarly we have 2sX[s] + s^2Y[s] = - 1/(s-2) --->(2) ...........from (1) and (2) we have s^2X[s] - 2sX[s] - 1/(s-2) = 1/(s-2) ....==> s(s-2)X[s] = 2/(s-2) ....==> X[s] = 2/s(s-2)^2 = 1/2s - 1/2(s-2) +1/(s-2)^2 ......Now apply inverse L.T. to get x(t) = 1/2 - e^2t / 2 + t e^2t ................... and we have Y[s] = -(s+2) / s^2(s-2)^2 = -3/4s -1/2s^2 + 3/4(s-2) - 1/(s-2)^2 ............apply inverse L.T. to get y(t) = -3/4 -t/2 + 3(e^2t) /4 - te^2t ..................

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