What is the solution for y^{''}-5y^'-6y=-3x^2 ?

The solution for y^{''}-5y^'-6y=-3x^2 is y=c_{1}e^{6x}+c_{2}e^{-x}+(x^2)/2-(5x)/6+31/36

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y^{''}-5y^'-6y=-3x^2

y^{^{''}} - 5 y^{^{'}} - 6 y = - 3 x^{2}

y = c_{1} e^{6 x} + c_{2} e^{- x} + \frac{x ^{2}}{2} - \frac{5 x}{6} + \frac{31}{36}

Solution steps

y^{''} (x) - 5 y^{'} (x) - 6 y = - 3 x^{2}

Solve linear ODE:

y = c_{1} e^{6 x} + c_{2} e^{- x} + \frac{x ^{2}}{2} - \frac{5 x}{6} + \frac{31}{36}

y = c_{1} e^{6 x} + c_{2} e^{- x} + \frac{x ^{2}}{2} - \frac{5 x}{6} + \frac{31}{36}

y^{^{''}} - 2 y^{^{'}} + y = e^{2 x}, y (0) = 2, y^{^{'}} (0) = 0

y^{^{''}} + 4 y^{^{'}} + 9 y = x^{2} + 3 x

- y^{^{''}} + y^{^{'}} + 2 y = 72 t + 400 sin (t) + 36

y^{^{''}} - y^{^{'}} + y = 9 sin (t) - 21 e^{2 t}

y^{^{''}} - y^{^{'}} = 9 + e^{x}

What is the solution for y^{''}-5y^'-6y=-3x^2 ?
The solution for y^{''}-5y^'-6y=-3x^2 is y=c_{1}e^{6x}+c_{2}e^{-x}+(x^2)/2-(5x)/6+31/36