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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 2 y^{^{'}} + y = 30 e^{t} t^{- 7}

y^{^{''}} + 4 y = 12 t

y^{^{''}} - y^{^{'}} + 2 y = 2 e^{2 t} + 5 t - 1

y^{^{''}} + y^{^{'}} + y = k

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

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