What is the solution for y^{''}+y^'=-4x+2 ?

The solution for y^{''}+y^'=-4x+2 is y=c_{1}+c_{2}e^{-x}-2x^2+6x

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

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

y = c_{1} + c_{2} e^{- x} - 2 x^{2} + 6 x

Solution steps

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

Solve linear ODE:

y = c_{1} + c_{2} e^{- x} - 2 x^{2} + 6 x

y = c_{1} + c_{2} e^{- x} - 2 x^{2} + 6 x

y^{^{'''}} + 24 y^{^{''}} - 2048 y = e^{- 16 x}

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

y^{^{''}} - 2 y^{^{'}} + y = 8 e^{t}

y^{^{''}} + y = 2 sin (2 t), y (0) = 6, y^{^{'}} (0) = 0

d, d (t) = 12 t + 20

What is the solution for y^{''}+y^'=-4x+2 ?
The solution for y^{''}+y^'=-4x+2 is y=c_{1}+c_{2}e^{-x}-2x^2+6x