What is the solution for (D^2-5D+6)y=x^2e^{3x} ?

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 10 y^{^{'}} + 20 y = 50 sinh (5 x)

\frac{d ^{4} y}{d x ^{4}} + 3 \frac{d ^{2} y}{d x ^{2}} - 4 = 0

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

y^{^{''}} + 17 y = 5 e^{x}

y^{^{''}} + 3 y^{^{'}} + 2 y = \frac{1}{( 9 + e ^{x} )}

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