What is the solution for y^{''}+y^'-12y=-32t+24t^2 ?

The solution for y^{''}+y^'-12y=-32t+24t^2 is y=c_{1}e^{3t}+c_{2}e^{-4t}-2t^2+(7t)/3-5/36

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y^{''}+y^'-12y=-32t+24t^2

y^{^{''}} + y^{^{'}} - 12 y = - 32 t + 24 t^{2}

y = c_{1} e^{3 t} + c_{2} e^{- 4 t} - 2 t^{2} + \frac{7 t}{3} - \frac{5}{36}

Solution steps

y^{''} (t) + y^{'} (t) - 12 y = - 32 t + 24 t^{2}

Solve linear ODE:

y = c_{1} e^{3 t} + c_{2} e^{- 4 t} - 2 t^{2} + \frac{7 t}{3} - \frac{5}{36}

y = c_{1} e^{3 t} + c_{2} e^{- 4 t} - 2 t^{2} + \frac{7 t}{3} - \frac{5}{36}

\frac{d ^{2} y}{d x ^{2}} + y = 3 + e^{- x} + 5 e^{2 x}

y^{^{''}} + 20 y^{^{'}} + 200 y = 320

y^{^{'''}} - 2 y^{^{'}} + 4 y = sin (2 x)

y^{^{''}} + 9 y^{^{'}} + 14 y = 3 sin (2 x)

x^{^{''}} - x = e^{- t}

What is the solution for y^{''}+y^'-12y=-32t+24t^2 ?
The solution for y^{''}+y^'-12y=-32t+24t^2 is y=c_{1}e^{3t}+c_{2}e^{-4t}-2t^2+(7t)/3-5/36