What is the solution for y^{''}+y=-2x-3sin(2x) ?

The solution for y^{''}+y=-2x-3sin(2x) is y=c_{1}cos(x)+c_{2}sin(x)-2x+sin(2x)

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y^{''}+y=-2x-3sin(2x)

y^{^{''}} + y = - 2 x - 3 sin (2 x)

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

Solution steps

y^{''} (x) + y = - 2 x - 3 sin (2 x)

Solve linear ODE:

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

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

y^{^{''}} - 6 y^{^{'}} + 9 y = 3 e^{3 x} + 2 sin (x) - cos (x)

2 y^{^{''}} + 2 y = - 16 cos (3 x)

y^{^{'''}} - y^{^{''}} - y^{^{'}} + y = 3 e^{- t} + 2

y^{^{''}} - 3 y^{^{'}} - 4 y = e^{2 x}

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

What is the solution for y^{''}+y=-2x-3sin(2x) ?
The solution for y^{''}+y=-2x-3sin(2x) is y=c_{1}cos(x)+c_{2}sin(x)-2x+sin(2x)