What is the solution for (D^2-6D+9)y=1+x+x^2 ?

The solution for (D^2-6D+9)y=1+x+x^2 is y=c_{1}e^{3x}+c_{2}xe^{3x}+(x^2)/9+(7x)/(27)+7/27

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(D^2-6D+9)y=1+x+x^2

(D^{2} - 6 D + 9) y = 1 + x + x^{2}

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + \frac{x ^{2}}{9} + \frac{7 x}{27} + \frac{7}{27}

Solution steps

(D^{2} - 6 D + 9) y = 1 + x + x^{2}

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + \frac{x ^{2}}{9} + \frac{7 x}{27} + \frac{7}{27}

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + \frac{x ^{2}}{9} + \frac{7 x}{27} + \frac{7}{27}

y^{^{''}} + 3 y^{^{'}} + 13 y = 52, y (0) = 2, y^{^{'}} (0) = 1

y^{^{''}} + 2 y^{^{'}} - 3 y = 5 e^{2 x}, y (0) = 0, y^{^{'}} (0) = 5

y^{^{''}} + 3 y^{^{'}} + 2 y = 10 e^{x}

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

y^{^{''}} + 8 y^{^{'}} + 15 y = 10, y (0) = 2, y^{^{'}} (0) = 1

What is the solution for (D^2-6D+9)y=1+x+x^2 ?
The solution for (D^2-6D+9)y=1+x+x^2 is y=c_{1}e^{3x}+c_{2}xe^{3x}+(x^2)/9+(7x)/(27)+7/27