What is the solution for linear x^{''}+x^'=1-2t+t^2 ?

The solution for linear x^{''}+x^'=1-2t+t^2 is x=c_{1}+c_{2}e^{-t}+(t^3)/3-2t^2+5t

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linear x^{''}+x^'=1-2t+t^2

linear

x^{^{''}} + x^{^{'}} = 1 - 2 t + t^{2}

x = c_{1} + c_{2} e^{- t} + \frac{t ^{3}}{3} - 2 t^{2} + 5 t

Solution steps

x^{''} (t) + x^{'} (t) = 1 - 2 t + t^{2}

Solve linear ODE:

x = c_{1} + c_{2} e^{- t} + \frac{t ^{3}}{3} - 2 t^{2} + 5 t

x = c_{1} + c_{2} e^{- t} + \frac{t ^{3}}{3} - 2 t^{2} + 5 t

y^{^{''}} + 25 y = 7 x

y^{^{''}} + y^{^{'}} - 6 y = 20 e^{t}, y (0) = 0, y^{^{'}} (0) = 1

2 y^{^{''}} + 6 y^{^{'}} = 10 sin (7 t)

y^{^{''}} + 4 y = 16 t^{3} sin (2 t)

y^{^{''}} + 11 y^{^{'}} + 18 y = 5

What is the solution for linear x^{''}+x^'=1-2t+t^2 ?
The solution for linear x^{''}+x^'=1-2t+t^2 is x=c_{1}+c_{2}e^{-t}+(t^3)/3-2t^2+5t