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

The solution for y^{''}-3y^'=6xe^{3x} is y=c_{1}e^{3x}+c_{2}+e^{3x}x^2-2/9 (3e^{3x}x-e^{3x})

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y^{''}-3y^'=6xe^{3x}

y^{^{''}} - 3 y^{^{'}} = 6 x e^{3 x}

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

Solution steps

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

Solve linear ODE:

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

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

What is the solution for y^{''}-3y^'=6xe^{3x} ?
The solution for y^{''}-3y^'=6xe^{3x} is y=c_{1}e^{3x}+c_{2}+e^{3x}x^2-2/9 (3e^{3x}x-e^{3x})