What is the solution for y^{''}+5y^'+4y=e^{-x}+2 ?

The solution for y^{''}+5y^'+4y=e^{-x}+2 is y=c_{1}e^{-x}+c_{2}e^{-4x}+(e^{-x}x)/3+1/2

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y^{''}+5y^'+4y=e^{-x}+2

y^{^{''}} + 5 y^{^{'}} + 4 y = e^{- x} + 2

y = c_{1} e^{- x} + c_{2} e^{- 4 x} + \frac{e ^{- x} x}{3} + \frac{1}{2}

Solution steps

y^{''} (x) + 5 y^{'} (x) + 4 y = e^{- x} + 2

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} e^{- 4 x} + \frac{e ^{- x} x}{3} + \frac{1}{2}

y = c_{1} e^{- x} + c_{2} e^{- 4 x} + \frac{e ^{- x} x}{3} + \frac{1}{2}

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

y^{^{''}} + 8 y^{^{'}} + 25 y = 150, y^{^{'}} (0) = 0, y (0) = 0

y^{^{''}} - 5 y^{^{'}} + 4 y = 2 t e^{t}, y (0) = 1, y^{^{'}} (0) = - 4

\frac{d ^{2} y}{d x ^{2}} + 14 \frac{d y}{d x} + 49 y = 4 e^{3 x}

y^{^{''}} - 2 y^{^{'}} - 3 y = 12 e x - 12

What is the solution for y^{''}+5y^'+4y=e^{-x}+2 ?
The solution for y^{''}+5y^'+4y=e^{-x}+2 is y=c_{1}e^{-x}+c_{2}e^{-4x}+(e^{-x}x)/3+1/2