\(電腦在工程上的應用\ \ \ \ \ \ 第八次作業\ \ \ \ \ \ \ \ \ 河4B\ \ 陳思瑋\ \ B94520155\)
eq1 = y''[t] + y[t] == Cos[0.5  t]
LaplaceTransform[eq1, t, s]

InverseLaplaceTransform[
      (s/(0.25 + (s^2)))*(1/(1 + (s^2))), s, 
      t]

Plot[
      y[t] = (1.3333333333333333` + 
                0.` I) Cos[
              0.5` t] - (1.3333333333333333` + 0.` I) Cos[
              1.` t] + (0.` + 
                0.` I) Sin[
              0.5` t] + (0.` + 
                0.` I) Sin[1.` t], {t, 0, 
        15  π}]

eq2 = y''[t] + y[t] == 
        Cos[0.99  t]
LaplaceTransform[eq2, t, s]

InverseLaplaceTransform[
      (s/(0.9801` + (s^2)))*(1/(1 + (s^2))), s, 
      t]

Plot[
      y[t] = (50.25125628140696` + 
                0.` I) Cos[
              0.99` t] - (50.25125628140696` + 0.` I) Cos[
              1.` t] + (0.` + 
                0.` I) Sin[
              0.99` t] + (0.` + 
                0.` I) Sin[1.` t], {t, 0, 
        200  π}]

eq3 = y''[t] + y[t] == 
        Cos[1.01  t]
LaplaceTransform[eq3, t, s]

InverseLaplaceTransform[
      (s/(1.0201` + (s^2)))*(1/(1 + (s^2))), s, 
      t]

Plot[
      y[t] = (49.75124378109451` + 
                0.` I) Cos[
              1.` t] - (49.75124378109451` + 
                0.` I) Cos[
              1.01` t] + (0.` + 
                0.` I) Sin[
              1.` t] + (0.` + 
                0.` I) Sin[1.01` t], {t, 0, 
        200  π}]

eq4 = y''[t] + y[t] == Cos[t]
LaplaceTransform[eq4, t, s]

InverseLaplaceTransform[(s/(1 + (s^2)))*(1/(1 + (s^2))), s, 
      t]

Plot[y[t] = (1/2) t Sin[t], {t, 0, 400  π}]

(t^2) y''[t] + ty'[t] == 0
A = D[D[(s^2) y[s] - sy[0] - y'[0], s], s]

B = -1*D[s*y[s] - y[0], s]

A + B

c = D[D[(v^2) y[v] - vy[0] - y'[0], v], v]

d = -1*D[v*y[v] - y[0], v]*3

c + d + y[v] // Simplify

v (y'[v]+v y''[v])    又跟原方程式一樣