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

The solution for y^{''}+y^'-6y=x^2+3 is y=c_{1}e^{2x}+c_{2}e^{-3x}-(x^2)/6-x/(18)-61/108

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

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

y = c_{1} e^{2 x} + c_{2} e^{- 3 x} - \frac{x ^{2}}{6} - \frac{x}{18} - \frac{61}{108}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- 3 x} - \frac{x ^{2}}{6} - \frac{x}{18} - \frac{61}{108}

y = c_{1} e^{2 x} + c_{2} e^{- 3 x} - \frac{x ^{2}}{6} - \frac{x}{18} - \frac{61}{108}

y^{^{''}} + 4 y^{^{'}} + 4 y = 6 sin (x)

y^{^{'''}} - y^{^{'}} = sin^{2} (x)

y^{^{''}} + y^{^{'}} = csc (x)

y^{^{''}} + 9 y = 2, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + 2 y^{^{'}} + y = t

What is the solution for y^{''}+y^'-6y=x^2+3 ?
The solution for y^{''}+y^'-6y=x^2+3 is y=c_{1}e^{2x}+c_{2}e^{-3x}-(x^2)/6-x/(18)-61/108