x^{''}+2x^'+x=2te^{-t}

Lösung

x^{^{''}} + 2 x^{^{'}} + x = 2 t e^{- t}

x = c_{1} e^{- t} + c_{2} t e^{- t} + \frac{e ^{- t} t ^{3}}{3}

Schritte zur Lösung

x^{''} (t) + 2 x^{'} (t) + x = 2 t e^{- t}

Solve linear ODE:

x = c_{1} e^{- t} + c_{2} t e^{- t} + \frac{e ^{- t} t ^{3}}{3}

x = c_{1} e^{- t} + c_{2} t e^{- t} + \frac{e ^{- t} t ^{3}}{3}

y^{^{''}} + y = 2 t

\frac{1}{6} y^{^{''}} + 6 y = tan (6 t) - \frac{1}{2} e^{t}

2 y^{^{''}} + 3 y^{^{'}} - 5 y = 2 cos (x)

y^{^{''}} - 4 y = e^{- x}

y^{^{''}} - 4 y^{^{'}} + 9 y = 2 x - 3