What is the solution for y^{'''}+3y^{''}+3y^'+y=e-x ?

The solution for y^{'''}+3y^{''}+3y^'+y=e-x is y=c_{1}e^{-x}+c_{2}xe^{-x}+c_{3}x^2e^{-x}-x+e+3

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y^{'''}+3y^{''}+3y^'+y=e-x

y^{^{'''}} + 3 y^{^{''}} + 3 y^{^{'}} + y = e - x

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

Solution steps

y^{'''} (x) + 3 y^{''} (x) + 3 y^{'} (x) + y = e - x

Solve linear ODE:

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

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

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