What is the solution for y^{''}+y^'-42y=-130t+84t^2 ?

The solution for y^{''}+y^'-42y=-130t+84t^2 is y=c_{1}e^{6t}+c_{2}e^{-7t}-2t^2+3t-1/42

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y^{''}+y^'-42y=-130t+84t^2

y^{^{''}} + y^{^{'}} - 42 y = - 130 t + 84 t^{2}

y = c_{1} e^{6 t} + c_{2} e^{- 7 t} - 2 t^{2} + 3 t - \frac{1}{42}

Solution steps

y^{''} (t) + y^{'} (t) - 42 y = - 130 t + 84 t^{2}

Solve linear ODE:

y = c_{1} e^{6 t} + c_{2} e^{- 7 t} - 2 t^{2} + 3 t - \frac{1}{42}

y = c_{1} e^{6 t} + c_{2} e^{- 7 t} - 2 t^{2} + 3 t - \frac{1}{42}

\frac{d ^{2} y}{d t ^{2}} + 12 \frac{d y}{d t} + 32 y = 32 H (t)

d, d (t) = - t^{2} + 5 t

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

y^{^{''}} + 3 y^{^{'}} - 10 = 0, y (0) = 1, y^{^{'}} (0) = 9

y^{^{''}} + y^{^{'}} - 12 y = e^{2 x}

What is the solution for y^{''}+y^'-42y=-130t+84t^2 ?
The solution for y^{''}+y^'-42y=-130t+84t^2 is y=c_{1}e^{6t}+c_{2}e^{-7t}-2t^2+3t-1/42