What is the solution for y^{''}-9y=-27t+18t^2 ?

The solution for y^{''}-9y=-27t+18t^2 is y=c_{1}e^{3t}+c_{2}e^{-3t}-2t^2+3t-4/9

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y^{''}-9y=-27t+18t^2

y^{^{''}} - 9 y = - 27 t + 18 t^{2}

y = c_{1} e^{3 t} + c_{2} e^{- 3 t} - 2 t^{2} + 3 t - \frac{4}{9}

Solution steps

y^{''} (t) - 9 y = - 27 t + 18 t^{2}

Solve linear ODE:

y = c_{1} e^{3 t} + c_{2} e^{- 3 t} - 2 t^{2} + 3 t - \frac{4}{9}

y = c_{1} e^{3 t} + c_{2} e^{- 3 t} - 2 t^{2} + 3 t - \frac{4}{9}

y^{^{'''}} - 2 y^{^{''}} + 26 y^{^{'}} = 174 e^{3 x}

y^{^{''}} + y^{^{'}} - y = t^{3}, y (0) = 1, y^{^{'}} (0) = 0

- y^{^{''}} + 2 y^{^{'}} + 3 y = 1, y (0) = 0, y (1) = (0)

y^{^{''}} - 3 y = e^{- 2 x}

y^{^{''}} - 2 y^{^{'}} + y = \frac{e ^{- x}}{1 + x ^{2}}

What is the solution for y^{''}-9y=-27t+18t^2 ?
The solution for y^{''}-9y=-27t+18t^2 is y=c_{1}e^{3t}+c_{2}e^{-3t}-2t^2+3t-4/9