What is the solution for y^{''}-4y^'+4y=e^{2x}(12x^2-6x) ?

The solution for y^{''}-4y^'+4y=e^{2x}(12x^2-6x) is y=c_{1}e^{2x}+c_{2}xe^{2x}+e^{2x}x^4-e^{2x}x^3

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y^{''}-4y^'+4y=e^{2x}(12x^2-6x)

y^{^{''}} - 4 y^{^{'}} + 4 y = e^{2 x} (12 x^{2} - 6 x)

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

Solution steps

y^{''} (x) - 4 y^{'} (x) + 4 y = e^{2 x} (12 x^{2} - 6 x)

Solve linear ODE:

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

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

y^{^{'''}} - 3 y^{^{''}} + 3 y^{^{'}} = x - 8 e^{x}

y^{^{'''}} - y^{^{''}} - 16 y^{^{'}} + 16 y = 9 - e^{x} + e^{4 x}

y^{^{''}} + y = t^{2} + 2

y^{^{''}} - 16 y = 6

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

What is the solution for y^{''}-4y^'+4y=e^{2x}(12x^2-6x) ?
The solution for y^{''}-4y^'+4y=e^{2x}(12x^2-6x) is y=c_{1}e^{2x}+c_{2}xe^{2x}+e^{2x}x^4-e^{2x}x^3