y^{''}+10y^'+41y=te^{-5t}sin(4t)

Lösung

y^{^{''}} + 10 y^{^{'}} + 41 y = t e^{- 5 t} sin (4 t)

y = e^{- 5 t} (c_{1} cos (4 t) + c_{2} sin (4 t)) + \frac{- e ^{- 5 t} cos ( 4 t ) ( 32 t ^{2} - 8 t sin ( 8 t ) - cos ( 8 t ) ) + e ^{- 5 t} sin ( 4 t ) ( - 16 t cos ^{2} ( 4 t ) + 8 t + sin ( 8 t ) )}{512}

Schritte zur Lösung

y^{''} (t) + 10 y^{'} (t) + 41 y = t e^{- 5 t} sin (4 t)

Solve linear ODE:

y = e^{- 5 t} (c_{1} cos (4 t) + c_{2} sin (4 t)) + \frac{- e ^{- 5 t} cos ( 4 t ) ( 32 t ^{2} - 8 t sin ( 8 t ) - cos ( 8 t ) ) + e ^{- 5 t} sin ( 4 t ) ( - 16 t cos ^{2} ( 4 t ) + 8 t + sin ( 8 t ) )}{512}

y = e^{- 5 t} (c_{1} cos (4 t) + c_{2} sin (4 t)) + \frac{- e ^{- 5 t} cos ( 4 t ) ( 32 t ^{2} - 8 t sin ( 8 t ) - cos ( 8 t ) ) + e ^{- 5 t} sin ( 4 t ) ( - 16 t cos ^{2} ( 4 t ) + 8 t + sin ( 8 t ) )}{512}

y^{^{''}} - y^{^{'}} - 6 y = e^{t}, y (0) = 1, y^{^{'}} (0) = 1

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

y^{^{''}} + 3 y^{^{'}} + 2 y = e^{x}, y (0) = 0, y^{^{'}} (0) = 3

y^{^{''}} - y^{^{'}} + \frac{y}{4} = 3 + e^{\frac{x}{2}}

y^{^{''}} + 10 y^{^{'}} + 16 y = 40 cos (4 x)