What is the solution for y^{''}-36y=-108t+72t^2 ?

The solution for y^{''}-36y=-108t+72t^2 is y=c_{1}e^{6t}+c_{2}e^{-6t}-2t^2+3t-1/9

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y^{''}-36y=-108t+72t^2

y^{^{''}} - 36 y = - 108 t + 72 t^{2}

y = c_{1} e^{6 t} + c_{2} e^{- 6 t} - 2 t^{2} + 3 t - \frac{1}{9}

Solution steps

y^{''} (t) - 36 y = - 108 t + 72 t^{2}

Solve linear ODE:

y = c_{1} e^{6 t} + c_{2} e^{- 6 t} - 2 t^{2} + 3 t - \frac{1}{9}

y = c_{1} e^{6 t} + c_{2} e^{- 6 t} - 2 t^{2} + 3 t - \frac{1}{9}

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} - 3 y = 4 e^{x} - 9

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = cos (2 x)

x^{^{''}} + 2 x = cos (8 t)

y^{^{''}} + 12 y^{^{'}} + 37 = 0, y (0) = 1, y^{^{'}} (0) = 0

y^{^{''}} - y^{^{'}} = 2 t^{2} + t - 1

What is the solution for y^{''}-36y=-108t+72t^2 ?
The solution for y^{''}-36y=-108t+72t^2 is y=c_{1}e^{6t}+c_{2}e^{-6t}-2t^2+3t-1/9