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}

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

y^{^{''''}} - 2 y^{^{'''}} + y^{^{''}} = 3 x^{2} + 2 x - e^{x}

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

6 y^{^{''}} - 13 y^{^{'}} - 5 y = 5 x^{3} + 39 x^{2} - 36 x - 10

y^{^{''}} - 4 y = 4 t - 4

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