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

The solution for y^{''}+y^'=-2x-3sin(2x) is y=c_{1}+c_{2}e^{-x}-x^2+2x+3/5 sin(2x)+3/10 cos(2x)

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

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

y = c_{1} + c_{2} e^{- x} - x^{2} + 2 x + \frac{3}{5} sin (2 x) + \frac{3}{10} cos (2 x)

Solution steps

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

Solve linear ODE:

y = c_{1} + c_{2} e^{- x} - x^{2} + 2 x + \frac{3}{5} sin (2 x) + \frac{3}{10} cos (2 x)

y = c_{1} + c_{2} e^{- x} - x^{2} + 2 x + \frac{3}{5} sin (2 x) + \frac{3}{10} cos (2 x)

What is the solution for y^{''}+y^'=-2x-3sin(2x) ?
The solution for y^{''}+y^'=-2x-3sin(2x) is y=c_{1}+c_{2}e^{-x}-x^2+2x+3/5 sin(2x)+3/10 cos(2x)