What is the solution for y^{''}+y=8xcos(x) ?

The solution for y^{''}+y=8xcos(x) is y=c_{1}cos(x)+c_{2}sin(x)+2x^2sin(x)+2xcos(x)

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y^{''}+y=8xcos(x)

y^{^{''}} + y = 8 x cos (x)

y = c_{1} cos (x) + c_{2} sin (x) + 2 x^{2} sin (x) + 2 x cos (x)

Solution steps

y^{''} (x) + y = 8 x cos (x)

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) + 2 x^{2} sin (x) + 2 x cos (x)

y = c_{1} cos (x) + c_{2} sin (x) + 2 x^{2} sin (x) + 2 x cos (x)

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

y^{^{''}} + 2 y^{^{'}} + 10 = 0, y (0) = - 2, y^{^{'}} (0) = 0

4 y^{^{''}} + y = 10 cos (x)

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

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

What is the solution for y^{''}+y=8xcos(x) ?
The solution for y^{''}+y=8xcos(x) is y=c_{1}cos(x)+c_{2}sin(x)+2x^2sin(x)+2xcos(x)