What is the solution for 9y^{''}-6y^'+y=9-x^2 ?

The solution for 9y^{''}-6y^'+y=9-x^2 is y=c_{1}e^{x/3}+c_{2}e^{x/3}x-x^2-12x-45

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9y^{''}-6y^'+y=9-x^2

9 y^{^{''}} - 6 y^{^{'}} + y = 9 - x^{2}

y = c_{1} e^{\frac{x}{3}} + c_{2} e^{\frac{x}{3}} x - x^{2} - 12 x - 45

Solution steps

9 y^{''} (x) - 6 y^{'} (x) + y = 9 - x^{2}

Solve linear ODE:

y = c_{1} e^{\frac{x}{3}} + c_{2} e^{\frac{x}{3}} x - x^{2} - 12 x - 45

y = c_{1} e^{\frac{x}{3}} + c_{2} e^{\frac{x}{3}} x - x^{2} - 12 x - 45

y^{^{''}} - y^{^{'}} - 2 y = sin (3 t), y (0) = 1, y^{^{'}} (0) = - 1

y^{^{''}} - 2 y^{^{'}} + 5 y = 1 + t, y (0) = 0, y^{^{'}} (0) = 1

y^{^{''}} + y = cos (x) + 4

y^{^{''}} - 2 y^{^{'}} + 5 = 0

y^{^{''}} - 8 y^{^{'}} + 20 y = 200 x^{2} - 39

What is the solution for 9y^{''}-6y^'+y=9-x^2 ?
The solution for 9y^{''}-6y^'+y=9-x^2 is y=c_{1}e^{x/3}+c_{2}e^{x/3}x-x^2-12x-45