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

The solution for y^{'''}+y^{''}=x+e^{-x} is y=c_{1}e^{-x}+c_{2}+c_{3}x+(x^3)/6-(x^2)/2+e^{-x}x

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

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

y = c_{1} e^{- x} + c_{2} + c_{3} x + \frac{x ^{3}}{6} - \frac{x ^{2}}{2} + e^{- x} x

Solution steps

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

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} + c_{3} x + \frac{x ^{3}}{6} - \frac{x ^{2}}{2} + e^{- x} x

y = c_{1} e^{- x} + c_{2} + c_{3} x + \frac{x ^{3}}{6} - \frac{x ^{2}}{2} + e^{- x} x

y^{^{''}} - 2 y^{^{'}} + y = e^{2 x} + x + 1

y^{^{''}} - 7 y^{^{'}} + 6 y = (20 x + 11) e^{x}

y^{^{''}} - y^{^{'}} + \frac{1}{4} y = 7 + e^{\frac{x}{2}}

V^{^{''}} + 25 \cdot 1 0^{2} V^{^{'}} + 25 \cdot 1 0^{6} V = 5 \cdot 1 0^{8}

y^{^{''}} + 49 y = 6 cos (7 x) + 9 sin (7 x)

What is the solution for y^{'''}+y^{''}=x+e^{-x} ?
The solution for y^{'''}+y^{''}=x+e^{-x} is y=c_{1}e^{-x}+c_{2}+c_{3}x+(x^3)/6-(x^2)/2+e^{-x}x