What is the solution for y^{''}-6y^'+9y=3x^2+4e^{4x} ?

The solution for y^{''}-6y^'+9y=3x^2+4e^{4x} is y=c_{1}e^{3x}+c_{2}xe^{3x}+(x^2)/3+(4x+2)/9+4e^{4x}

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y^{''}-6y^'+9y=3x^2+4e^{4x}

y^{^{''}} - 6 y^{^{'}} + 9 y = 3 x^{2} + 4 e^{4 x}

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + \frac{x ^{2}}{3} + \frac{4 x + 2}{9} + 4 e^{4 x}

Solution steps

y^{''} (x) - 6 y^{'} (x) + 9 y = 3 x^{2} + 4 e^{4 x}

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + \frac{x ^{2}}{3} + \frac{4 x + 2}{9} + 4 e^{4 x}

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + \frac{x ^{2}}{3} + \frac{4 x + 2}{9} + 4 e^{4 x}

y^{^{''}} + 6 y = - 294 x^{2} e^{6} x

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

y^{^{''}} - y^{^{'}} - 2 y = 1

v^{^{''}} - 16 v = 2 e^{4 x}

y^{^{''}} + 2 y^{^{'}} + y = t^{5} e^{- t}

What is the solution for y^{''}-6y^'+9y=3x^2+4e^{4x} ?
The solution for y^{''}-6y^'+9y=3x^2+4e^{4x} is y=c_{1}e^{3x}+c_{2}xe^{3x}+(x^2)/3+(4x+2)/9+4e^{4x}