What is the solution for y^{''}-y^'-6y=-14t+12t^2 ?

The solution for y^{''}-y^'-6y=-14t+12t^2 is y=c_{1}e^{3t}+c_{2}e^{-2t}-2t^2+3t-7/6

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y^{''}-y^'-6y=-14t+12t^2

y^{^{''}} - y^{^{'}} - 6 y = - 14 t + 12 t^{2}

y = c_{1} e^{3 t} + c_{2} e^{- 2 t} - 2 t^{2} + 3 t - \frac{7}{6}

Solution steps

y^{''} (t) - y^{'} (t) - 6 y = - 14 t + 12 t^{2}

Solve linear ODE:

y = c_{1} e^{3 t} + c_{2} e^{- 2 t} - 2 t^{2} + 3 t - \frac{7}{6}

y = c_{1} e^{3 t} + c_{2} e^{- 2 t} - 2 t^{2} + 3 t - \frac{7}{6}

4 y^{^{''}} - 4 y^{^{'}} + y = 16 e \frac{t}{2}

(D^{2} - 2 D + 1) y = x e^{x} sin (x)

y^{^{''}} + 100 y = 200 sin (10 x), y^{^{'}} (0) = 0, y^{^{''}} (0) = 0

y^{^{''}} + 9 y = cos (3 x) + sin (3 x), y (0) = 2, y^{^{'}} (0) = 1

y^{^{''}} - 6 y^{^{'}} + 9 y = t^{- 6} e^{3 t}

What is the solution for y^{''}-y^'-6y=-14t+12t^2 ?
The solution for y^{''}-y^'-6y=-14t+12t^2 is y=c_{1}e^{3t}+c_{2}e^{-2t}-2t^2+3t-7/6