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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 4 y^{^{'}} + 4 y = x^{2} + 3

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

9 y^{^{''}} + 4 y = 3 x + 12 e^{- x}, y (0) = 1, y^{^{'}} (0) = 0

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

4 y^{^{''}} - 3 y^{^{'}} = 4 x - 6

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