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

The solution for y^{''}-6y^'+9y=x^2+e^x is y=c_{1}e^{3x}+c_{2}xe^{3x}+(x^2)/9+(4x+2)/(27)+1/4 e^x

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

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

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + \frac{x ^{2}}{9} + \frac{4 x + 2}{27} + \frac{1}{4} e^{x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + \frac{x ^{2}}{9} + \frac{4 x + 2}{27} + \frac{1}{4} e^{x}

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + \frac{x ^{2}}{9} + \frac{4 x + 2}{27} + \frac{1}{4} e^{x}

D, D (t) = \frac{100 ln ( t + 2 )}{t + 2}

2 y^{^{''}} + 3 y^{^{'}} - 2 y = 6 x^{2} + 5 x - 8

y^{^{''}} + 4 y = 2 cos (2 x)

y^{^{''}} - 2 y^{^{'}} - 3 y = 12 e^{x} - 3

y^{^{''}} + y^{^{'}} + y = x sin (x)

What is the solution for y^{''}-6y^'+9y=x^2+e^x ?
The solution for y^{''}-6y^'+9y=x^2+e^x is y=c_{1}e^{3x}+c_{2}xe^{3x}+(x^2)/9+(4x+2)/(27)+1/4 e^x