(D^2+13D+13)x=12t^2+2t-cos(t)

Lösung

(D^{2} + 13 D + 13) x = 12 t^{2} + 2 t - cos (t)

x = c_{1} e^{\frac{( - 13 + 3 13 ) t}{2}} + c_{2} e^{\frac{( - 13 - 3 13 ) t}{2}} + \frac{12 t ^{2} - 22 t}{13} + \frac{262}{169} - \frac{13}{313} sin (t) - \frac{12}{313} cos (t)

Schritte zur Lösung

(D^{2} + 13 D + 13) x = 12 t^{2} + 2 t - cos (t)

Solve linear ODE:

x = c_{1} e^{\frac{( - 13 + 3 13 ) t}{2}} + c_{2} e^{\frac{( - 13 - 3 13 ) t}{2}} + \frac{12 t ^{2} - 22 t}{13} + \frac{262}{169} - \frac{13}{313} sin (t) - \frac{12}{313} cos (t)

x = c_{1} e^{\frac{( - 13 + 3 13 ) t}{2}} + c_{2} e^{\frac{( - 13 - 3 13 ) t}{2}} + \frac{12 t ^{2} - 22 t}{13} + \frac{262}{169} - \frac{13}{313} sin (t) - \frac{12}{313} cos (t)

y^{^{''}} - 6 y^{^{'}} + 9 y = t, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} - y = x e^{\frac{x}{2}}

y^{^{''}} + 9 y = e^{2 x} + x^{2} sin (x)

y^{^{''}} + 6 y^{^{'}} + 8 y = 2 t + e^{t}

x^{^{''}} + 25 x = 2 cos (5.1 t)