What is the solution for (d^2y)/(dx^2)+(dy)/(dx)4+8y=8x^2 ?

The solution for (d^2y)/(dx^2)+(dy)/(dx)4+8y=8x^2 is y=e^{-2x}(c_{1}cos(2x)+c_{2}sin(2x))+x^2-x+1/4

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(d^2y)/(dx^2)+(dy)/(dx)4+8y=8x^2

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} 4 + 8 y = 8 x^{2}

y = e^{- 2 x} (c_{1} cos (2 x) + c_{2} sin (2 x)) + x^{2} - x + \frac{1}{4}

Solution steps

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} 4 + 8 y = 8 x^{2}

Solve linear ODE:

y = e^{- 2 x} (c_{1} cos (2 x) + c_{2} sin (2 x)) + x^{2} - x + \frac{1}{4}

y = e^{- 2 x} (c_{1} cos (2 x) + c_{2} sin (2 x)) + x^{2} - x + \frac{1}{4}

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

\frac{1}{4} y^{^{''}} + y^{^{'}} + y = sin (x)

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

y^{^{''}} + y = sin (t), y (0) = (0), y^{^{'}} (0) = 1

y^{^{''}} - 7 y^{^{'}} + 12 y = - 288 t^{3}

What is the solution for (d^2y)/(dx^2)+(dy)/(dx)4+8y=8x^2 ?
The solution for (d^2y)/(dx^2)+(dy)/(dx)4+8y=8x^2 is y=e^{-2x}(c_{1}cos(2x)+c_{2}sin(2x))+x^2-x+1/4