What is the solution for y^{''}+5y^'+6y=x^2 ?

The solution for y^{''}+5y^'+6y=x^2 is y=c_{1}e^{-2x}+c_{2}e^{-3x}+(x^2)/6-(5x)/(18)+19/108

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y^{''}+5y^'+6y=x^2

y^{^{''}} + 5 y^{^{'}} + 6 y = x^{2}

y = c_{1} e^{- 2 x} + c_{2} e^{- 3 x} + \frac{x ^{2}}{6} - \frac{5 x}{18} + \frac{19}{108}

Solution steps

y^{''} (x) + 5 y^{'} (x) + 6 y = x^{2}

Solve linear ODE:

y = c_{1} e^{- 2 x} + c_{2} e^{- 3 x} + \frac{x ^{2}}{6} - \frac{5 x}{18} + \frac{19}{108}

y = c_{1} e^{- 2 x} + c_{2} e^{- 3 x} + \frac{x ^{2}}{6} - \frac{5 x}{18} + \frac{19}{108}

y^{^{''}} + 9 y = e^{t}

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

y^{^{''}} - 7 y^{^{'}} + a = 0, y (0) = 1, y^{^{'}} (0) = 10

y^{^{''}} + 4 y = - 6, y (\frac{π}{8}) = - \frac{1}{2}, y^{^{'}} (\frac{π}{8}) = 2

y^{^{''}} - 4 y^{^{'}} + 8 y = 8 x^{2} - 2

What is the solution for y^{''}+5y^'+6y=x^2 ?
The solution for y^{''}+5y^'+6y=x^2 is y=c_{1}e^{-2x}+c_{2}e^{-3x}+(x^2)/6-(5x)/(18)+19/108