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

The solution for y^{'''}+y^{''}=4x+e^{-x} is y=c_{1}e^{-x}+c_{2}+c_{3}x+(2x^3)/3-2x^2+e^{-x}x

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 6 y^{^{'}} + 9 y = - 12 e^{3 x}

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

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

y^{^{''}} - 2 y^{^{'}} - 3 y = 6 - 3 e^{- x} + 4 cos (3 x)

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

What is the solution for y^{'''}+y^{''}=4x+e^{-x} ?
The solution for y^{'''}+y^{''}=4x+e^{-x} is y=c_{1}e^{-x}+c_{2}+c_{3}x+(2x^3)/3-2x^2+e^{-x}x