What is the solution for y^{''}-4y=8x^2 ?

The solution for y^{''}-4y=8x^2 is y=c_{1}e^{2x}+c_{2}e^{-2x}-2x^2-1

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y^{''}-4y=8x^2

y^{^{''}} - 4 y = 8 x^{2}

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

Solution steps

y^{''} (x) - 4 y = 8 x^{2}

Solve linear ODE:

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

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

x^{^{''}} - 4 x^{^{'}} + 3 x = \frac{e ^{t}}{1 + e ^{t}}

y^{^{''}} - 9 y^{^{'}} + 18 y = 9 + 18 x

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

y^{^{''}} + 6 y^{^{'}} + 9 y = 3

y^{^{''}} - 16 y = 32 + 45 e^{x}, y (0) = 0, y^{^{'}} (0) = 1

What is the solution for y^{''}-4y=8x^2 ?
The solution for y^{''}-4y=8x^2 is y=c_{1}e^{2x}+c_{2}e^{-2x}-2x^2-1