y^{'''}+2y^{''}-y^'-2y=5\H(t)

Lösung

y^{^{'''}} + 2 y^{^{''}} - y^{^{'}} - 2 y = 5 H (t)

y = c_{1} e^{- 2 t} + c_{2} e^{- t} + c_{3} e^{t} + e^{- 2 t} \cdot \int \frac{5 H ( t )}{3 e ^{- 2 t}} d t + e^{- t} \cdot \int - \frac{5 e ^{t} H ( t )}{2} d t + e^{t} \cdot \int \frac{5 H ( t )}{6 e ^{t}} d t

Schritte zur Lösung

y^{'''} (t) + 2 y^{''} (t) - y^{'} (t) - 2 y = 5 H (t)

Solve linear ODE:

y = c_{1} e^{- 2 t} + c_{2} e^{- t} + c_{3} e^{t} + e^{- 2 t} \cdot \int \frac{5 H ( t )}{3 e ^{- 2 t}} d t + e^{- t} \cdot \int - \frac{5 e ^{t} H ( t )}{2} d t + e^{t} \cdot \int \frac{5 H ( t )}{6 e ^{t}} d t

y = c_{1} e^{- 2 t} + c_{2} e^{- t} + c_{3} e^{t} + e^{- 2 t} \cdot \int \frac{5 H ( t )}{3 e ^{- 2 t}} d t + e^{- t} \cdot \int - \frac{5 e ^{t} H ( t )}{2} d t + e^{t} \cdot \int \frac{5 H ( t )}{6 e ^{t}} d t

y^{^{''''}} + 3 y^{^{'''}} - 4 y^{^{''}} - 4 y^{^{'}} + \frac{1}{2} = 0

y^{^{''}} - 5 y^{^{'}} - 6 y = - 6 e^{t}

y^{^{''}} - 4 y = e^{2 x}, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} - 3 y^{^{'}} - 28 y = 1056 e^{- t}

y^{^{''}} + y = g (t), y (0) = 0, y^{^{'}} (0) = 0