What is the solution for 4y^{''}+y=e^{-2t}cos(t/2)+6tsin(t/2) ?

The solution for 4y^{''}+y=e^{-2t}cos(t/2)+6tsin(t/2) is

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4y^{''}+y=e^{-2t}cos(t/2)+6tsin(t/2)

4 y^{^{''}} + y = e^{- 2 t} cos (\frac{t}{2}) + 6 t sin (\frac{t}{2})

y = c_{1} cos (\frac{t}{2}) + c_{2} sin (\frac{t}{2}) - \frac{1}{40} e^{- 2 t} sin (\frac{t}{2}) + \frac{1}{20} e^{- 2 t} cos (\frac{t}{2}) - \frac{3 t ^{2} cos ( \frac{t}{2} )}{4} + \frac{3 t sin ( \frac{t}{2} )}{2}

Solution steps

4 y^{''} (t) + y = e^{- 2 t} cos (\frac{t}{2}) + 6 t sin (\frac{t}{2})

Solve linear ODE:

y = c_{1} cos (\frac{t}{2}) + c_{2} sin (\frac{t}{2}) - \frac{1}{40} e^{- 2 t} sin (\frac{t}{2}) + \frac{1}{20} e^{- 2 t} cos (\frac{t}{2}) - \frac{3 t ^{2} cos ( \frac{t}{2} )}{4} + \frac{3 t sin ( \frac{t}{2} )}{2}

y = c_{1} cos (\frac{t}{2}) + c_{2} sin (\frac{t}{2}) - \frac{1}{40} e^{- 2 t} sin (\frac{t}{2}) + \frac{1}{20} e^{- 2 t} cos (\frac{t}{2}) - \frac{3 t ^{2} cos ( \frac{t}{2} )}{4} + \frac{3 t sin ( \frac{t}{2} )}{2}

y^{^{''''}} - 2 y^{^{'''}} + y^{^{''}} = t^{3}

y^{^{''}} - y - x = 0

(D^{2} + 2) y = e^{x + 2}

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

y^{^{''}} + 2 y^{^{'}} + 2 y = (t - π), y (0) = 2, y^{^{'}} (0) = 1

What is the solution for 4y^{''}+y=e^{-2t}cos(t/2)+6tsin(t/2) ?
The solution for 4y^{''}+y=e^{-2t}cos(t/2)+6tsin(t/2) is