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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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