What is the solution for (D^2-2D+1)y=4x^2e^x ?

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

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(D^2-2D+1)y=4x^2e^x

(D^{2} - 2 D + 1) y = 4 x^{2} e^{x}

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

Solution steps

(D^{2} - 2 D + 1) y = 4 x^{2} e^{x}

Solve linear ODE:

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

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

x^{^{''}} + 4 x^{^{'}} + 3 x = 4 e^{- 2 t}, x (0) = 0, x^{^{'}} (0) = 0

4 y^{^{''}} - 20 y^{^{'}} + 25 y = 6 - 5 x + 50 x^{2}

y^{^{''}} - 4 y^{^{'}} = sin^{2} (x)

y^{^{''}} - 4 y^{^{'}} + 5 y = 5 x^{2} + 2 x

y^{^{''}} - y^{^{'}} + 4 = 3 x^{2} + 4 x

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