What is the solution for y^{''}+5y^'+6y=xe^x ?

The solution for y^{''}+5y^'+6y=xe^x is y=c_{1}e^{-2x}+c_{2}e^{-3x}+(e^xx)/(12)-7/144 e^x

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y^{''}+5y^'+6y=xe^x

y^{^{''}} + 5 y^{^{'}} + 6 y = x e^{x}

y = c_{1} e^{- 2 x} + c_{2} e^{- 3 x} + \frac{e ^{x} x}{12} - \frac{7}{144} e^{x}

Solution steps

y^{''} (x) + 5 y^{'} (x) + 6 y = x e^{x}

Solve linear ODE:

y = c_{1} e^{- 2 x} + c_{2} e^{- 3 x} + \frac{e ^{x} x}{12} - \frac{7}{144} e^{x}

y = c_{1} e^{- 2 x} + c_{2} e^{- 3 x} + \frac{e ^{x} x}{12} - \frac{7}{144} e^{x}

y^{^{''}} - 10 y^{^{'}} + 21 y = 205 x^{2} - 100 x + 10

- x^{^{''}} + x^{^{'}} = 6 e^{t} + 2 t - 2

y^{^{''}} + 5 y^{^{'}} + 6 y = 12 e^{- 5 t} + 12 e^{- t} + 24

\frac{d ^{2} y}{d x ^{2}} - y = cos (2 x)

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

What is the solution for y^{''}+5y^'+6y=xe^x ?
The solution for y^{''}+5y^'+6y=xe^x is y=c_{1}e^{-2x}+c_{2}e^{-3x}+(e^xx)/(12)-7/144 e^x