What is the solution for y^{''}-81y=-243t+162t^2 ?

The solution for y^{''}-81y=-243t+162t^2 is y=c_{1}e^{9t}+c_{2}e^{-9t}-2t^2+3t-4/81

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y^{''}-81y=-243t+162t^2

y^{^{''}} - 81 y = - 243 t + 162 t^{2}

y = c_{1} e^{9 t} + c_{2} e^{- 9 t} - 2 t^{2} + 3 t - \frac{4}{81}

Solution steps

y^{''} (t) - 81 y = - 243 t + 162 t^{2}

Solve linear ODE:

y = c_{1} e^{9 t} + c_{2} e^{- 9 t} - 2 t^{2} + 3 t - \frac{4}{81}

y = c_{1} e^{9 t} + c_{2} e^{- 9 t} - 2 t^{2} + 3 t - \frac{4}{81}

5 x^{^{''}} + 10 x^{^{'}} + 30 x = 300

y^{^{''}} + y^{^{'}} - 2 = 0

- y^{^{''}} + y^{^{'}} + 6 y = 1 - x

y^{''} - 3 y^{'} + 2 y = 1 1^{(e^{3} x)}

y^{^{''}} - 2 y^{^{'}} + y = \frac{3 e ^{t}}{t ^{2} + 1}

What is the solution for y^{''}-81y=-243t+162t^2 ?
The solution for y^{''}-81y=-243t+162t^2 is y=c_{1}e^{9t}+c_{2}e^{-9t}-2t^2+3t-4/81