What is the solution for x^{''}+2x^'+5x=12cos(2t)+3sin(2t) ?

The solution for x^{''}+2x^'+5x=12cos(2t)+3sin(2t) is x=e^{-t}(c_{1}cos(2t)+c_{2}sin(2t))+3sin(2t)

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x^{''}+2x^'+5x=12cos(2t)+3sin(2t)

x^{^{''}} + 2 x^{^{'}} + 5 x = 12 cos (2 t) + 3 sin (2 t)

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

Solution steps

x^{''} (t) + 2 x^{'} (t) + 5 x = 12 cos (2 t) + 3 sin (2 t)

Solve linear ODE:

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

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

y^{^{''}} + 3 y^{^{'}} - 4 y = 2 cos (x)

y^{^{'''}} - y^{^{''}} - y^{^{'}} + y = 3 e^{- t} + 4

y^{^{'''}} - y^{^{''}} - y^{^{'}} + y = 3 e^{- t} + 5

- 4 y^{^{''}} + 9 y = 15

y^{^{''}} - 4 y^{^{'}} + 3 y = t^{2} e^{2 t}

What is the solution for x^{''}+2x^'+5x=12cos(2t)+3sin(2t) ?
The solution for x^{''}+2x^'+5x=12cos(2t)+3sin(2t) is x=e^{-t}(c_{1}cos(2t)+c_{2}sin(2t))+3sin(2t)