(d^2y)/(dt^2)+12(dy)/(dt)+32y=32\H(t)

Lösung

\frac{d ^{2} y}{d t ^{2}} + 12 \frac{d y}{d t} + 32 y = 32 H (t)

y = c_{1} e^{- 4 t} + c_{2} e^{- 8 t} + e^{- 4 t} \cdot \int 8 e^{4 t} H (t) d t + e^{- 8 t} \cdot \int - 8 e^{8 t} H (t) d t

Schritte zur Lösung

\frac{d ^{2} y}{d t ^{2}} + 12 \frac{d y}{d t} + 32 y = 32 H (t)

Solve linear ODE:

y = c_{1} e^{- 4 t} + c_{2} e^{- 8 t} + e^{- 4 t} \cdot \int 8 e^{4 t} H (t) d t + e^{- 8 t} \cdot \int - 8 e^{8 t} H (t) d t

y = c_{1} e^{- 4 t} + c_{2} e^{- 8 t} + e^{- 4 t} \cdot \int 8 e^{4 t} H (t) d t + e^{- 8 t} \cdot \int - 8 e^{8 t} H (t) d t

so l v e f or d, d (t) = - t^{2} + 5 t

y^{^{''}} + 2 y^{^{'}} + 2 y = 2 (x + 1)^{2}

y^{^{''}} + 3 y^{^{'}} - 10 = 0, y (0) = 1, y^{^{'}} (0) = 9

y^{^{''}} + y^{^{'}} - 12 y = e^{2 x}

y^{^{''}} + 4 y^{^{'}} + 50 y = 65 cos (2 t)