What is the solution for y^{''}-y^'-12y=2t^6e^{-3t} ?

The solution for y^{''}-y^'-12y=2t^6e^{-3t} is

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y^{''}-y^'-12y=2t^6e^{-3t}

y^{^{''}} - y^{^{'}} - 12 y = 2 t^{6} e^{- 3 t}

y = c_{1} e^{4 t} + c_{2} e^{- 3 t} - \frac{2 e ^{- 3 t} t ^{7}}{49} - \frac{2 e ^{- 3 t} t ^{6}}{49} - \frac{12 e ^{- 3 t} t ^{5}}{343} - \frac{60 e ^{- 3 t} t ^{4}}{2401} - \frac{240 e ^{- 3 t} t ^{3}}{16807} - \frac{720 e ^{- 3 t} t ^{2}}{117649} - \frac{1440 e ^{- 3 t} t}{823543}

Solution steps

y^{''} (t) - y^{'} (t) - 12 y = 2 t^{6} e^{- 3 t}

Solve linear ODE:

y = c_{1} e^{4 t} + c_{2} e^{- 3 t} - \frac{2 e ^{- 3 t} t ^{7}}{49} - \frac{2 e ^{- 3 t} t ^{6}}{49} - \frac{12 e ^{- 3 t} t ^{5}}{343} - \frac{60 e ^{- 3 t} t ^{4}}{2401} - \frac{240 e ^{- 3 t} t ^{3}}{16807} - \frac{720 e ^{- 3 t} t ^{2}}{117649} - \frac{1440 e ^{- 3 t} t}{823543}

y = c_{1} e^{4 t} + c_{2} e^{- 3 t} - \frac{2 e ^{- 3 t} t ^{7}}{49} - \frac{2 e ^{- 3 t} t ^{6}}{49} - \frac{12 e ^{- 3 t} t ^{5}}{343} - \frac{60 e ^{- 3 t} t ^{4}}{2401} - \frac{240 e ^{- 3 t} t ^{3}}{16807} - \frac{720 e ^{- 3 t} t ^{2}}{117649} - \frac{1440 e ^{- 3 t} t}{823543}

y^{^{''}} + y = 4, y (0) = 0, y^{^{'}} (0) = 0

- 3 y^{^{'''}} + y^{^{''''}} = 27 c e^{3 x}

4 q^{^{''}} + 8 q^{^{'}} + \frac{1}{0.0125} q = 88 t

x, d (x) = \frac{1}{2} g x^{2}

x^{^{''}} + x^{^{'}} + 2 x = cos (2 t)

What is the solution for y^{''}-y^'-12y=2t^6e^{-3t} ?
The solution for y^{''}-y^'-12y=2t^6e^{-3t} is