What is the solution for y^{''}+3y^'+2y=1+3x+x^2 ?

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

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y^{''}+3y^'+2y=1+3x+x^2

y^{^{''}} + 3 y^{^{'}} + 2 y = 1 + 3 x + x^{2}

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

Solution steps

y^{''} (x) + 3 y^{'} (x) + 2 y = 1 + 3 x + x^{2}

Solve linear ODE:

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

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

y^{^{''}} - y = e^{2 t} t^{2}

y^{^{''}} + y^{^{'}} - 2 y = 6

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

y^{^{''}} + 3 y^{^{'}} + 5 y = 4 t + 2 + e^{7 t}

y^{^{'''}} - y^{^{''}} - 16 y^{^{'}} + 16 y = 3 - e^{x} + e^{4 x}

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