What is the solution for y^{''}-5y^'+6y=2xe^{3x} ?

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

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y^{''}-5y^'+6y=2xe^{3x}

y^{^{''}} - 5 y^{^{'}} + 6 y = 2 x e^{3 x}

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 4 y^{^{'}} + 3 y = 1

y^{^{''}} + 9 y = 2 cos (3 x) + 2 sin (3 x)

y^{^{''}} - 32 y = 16 cos (x)

y^{^{''}} - 8 y^{^{'}} + 20 y = 200 x^{2} - 91

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

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