What is the solution for y^{''}-y^'-42y=-122t+84t^2 ?

The solution for y^{''}-y^'-42y=-122t+84t^2 is y=c_{1}e^{7t}+c_{2}e^{-6t}-2t^2+3t-1/6

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y^{''}-y^'-42y=-122t+84t^2

y^{^{''}} - y^{^{'}} - 42 y = - 122 t + 84 t^{2}

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

Solution steps

y^{''} (t) - y^{'} (t) - 42 y = - 122 t + 84 t^{2}

Solve linear ODE:

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

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

\frac{d ^{2} y}{d t ^{2}} + 12 \frac{d y}{d t} + 36 y = - 4 t + 3

y^{^{''}} - 2 y^{^{'}} = 4 x + e^{x}

y^{^{''}} + 2 y^{^{'}} = cos (t), y (0) = 0

y^{^{''}} + 7 y^{^{'}} = cos (2 x) + 8 e^{- 5 x}

y^{^{''}} + 3 y^{^{'}} - 4 y = 2, y^{^{'}} (0) = 0, y (0) = 0

What is the solution for y^{''}-y^'-42y=-122t+84t^2 ?
The solution for y^{''}-y^'-42y=-122t+84t^2 is y=c_{1}e^{7t}+c_{2}e^{-6t}-2t^2+3t-1/6