What is the solution for y^{''}+2y^'+5y=12cos(2x)+3sin(2x) ?

The solution for y^{''}+2y^'+5y=12cos(2x)+3sin(2x) is y=e^{-x}(c_{1}cos(2x)+c_{2}sin(2x))+3sin(2x)

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y^{''}+2y^'+5y=12cos(2x)+3sin(2x)

y^{^{''}} + 2 y^{^{'}} + 5 y = 12 cos (2 x) + 3 sin (2 x)

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

Solution steps

y^{''} (x) + 2 y^{'} (x) + 5 y = 12 cos (2 x) + 3 sin (2 x)

Solve linear ODE:

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

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

y^{^{''}} + 4 y = 2 x, y (0) = 1, y^{^{'}} (0) = 2

\frac{d ^{2} u}{d x ^{2}} - u = 6

y^{^{'''}} - 6 y^{^{''}} + 11 y^{^{'}} - 6 y = e^{x}

y^{^{''}} - 4 y = - 7 e^{2 x} + x

y^{^{''}} - 2 y^{^{'}} + 5 y = 1 + t

What is the solution for y^{''}+2y^'+5y=12cos(2x)+3sin(2x) ?
The solution for y^{''}+2y^'+5y=12cos(2x)+3sin(2x) is y=e^{-x}(c_{1}cos(2x)+c_{2}sin(2x))+3sin(2x)