y^{''}+4y^'+3y=1-\H(t-2)+\H(t-6)

Lösung

y^{^{''}} + 4 y^{^{'}} + 3 y = 1 - H (t - 2) + H (t - 6)

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

Schritte zur Lösung

y^{''} (t) + 4 y^{'} (t) + 3 y = 1 - H (t - 2) + H (t - 6)

Solve linear ODE:

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

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

2 y^{^{''}} + y^{^{'}} - 3 y = e^{x} - x

y^{^{''}} + 4 y = \frac{1}{sin ( 2 t )}

y^{^{''}} - 4 y^{^{'}} + 4 y = - 7 x + 4 x^{2}

0.5 y^{^{''}} + y^{^{'}} + 4 y = 10 sin (2 x)

y^{^{''}} - y^{^{'}} = \frac{e ^{x}}{e ^{x} + 1}