What is the solution for (D^2-2D+1)y=x^2e^{3x} ?

The solution for (D^2-2D+1)y=x^2e^{3x} is y=c_{1}e^x+c_{2}xe^x+(e^{3x}x^2)/4-(e^{3x}x)/2+3/8 e^{3x}

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(D^2-2D+1)y=x^2e^{3x}

(D^{2} - 2 D + 1) y = x^{2} e^{3 x}

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{3 x} x ^{2}}{4} - \frac{e ^{3 x} x}{2} + \frac{3}{8} e^{3 x}

Solution steps

(D^{2} - 2 D + 1) y = x^{2} e^{3 x}

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{3 x} x ^{2}}{4} - \frac{e ^{3 x} x}{2} + \frac{3}{8} e^{3 x}

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{3 x} x ^{2}}{4} - \frac{e ^{3 x} x}{2} + \frac{3}{8} e^{3 x}

What is the solution for (D^2-2D+1)y=x^2e^{3x} ?
The solution for (D^2-2D+1)y=x^2e^{3x} is y=c_{1}e^x+c_{2}xe^x+(e^{3x}x^2)/4-(e^{3x}x)/2+3/8 e^{3x}