x^{''}+4x^'+5x=e^{-2}tsin(t)

Lösung

x^{^{''}} + 4 x^{^{'}} + 5 x = e^{- 2} t sin (t)

x = e^{- 2 t} (c_{1} cos (t) + c_{2} sin (t)) + \frac{t sin ( t )}{8 e ^{2}} - \frac{t cos ( t )}{8 e ^{2}} - \frac{1}{16 e ^{2}} sin (t) + \frac{1}{8 e ^{2}} cos (t)

Schritte zur Lösung

x^{''} (t) + 4 x^{'} (t) + 5 x = e^{- 2} t sin (t)

Solve linear ODE:

x = e^{- 2 t} (c_{1} cos (t) + c_{2} sin (t)) + \frac{t sin ( t )}{8 e ^{2}} - \frac{t cos ( t )}{8 e ^{2}} - \frac{1}{16 e ^{2}} sin (t) + \frac{1}{8 e ^{2}} cos (t)

x = e^{- 2 t} (c_{1} cos (t) + c_{2} sin (t)) + \frac{t sin ( t )}{8 e ^{2}} - \frac{t cos ( t )}{8 e ^{2}} - \frac{1}{16 e ^{2}} sin (t) + \frac{1}{8 e ^{2}} cos (t)

so l v e f or d, d (t) = - 16 t^{2} + 32 t + 40

x^{^{''}} + 81 x = sin (7 t)

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

y^{^{''}} + 4 y^{^{'}} + 5 y = 2 cos (x) - 2 sin (x)

y^{^{''}} - y = \frac{( 4 e ^{- x} )}{( 1 - e ^{- 2 x} )}