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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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