What is the solution for y^{''}+6y^'+13y=30cos(x) ?

The solution for y^{''}+6y^'+13y=30cos(x) is y=e^{-3x}(c_{1}cos(2x)+c_{2}sin(2x))+sin(x)+2cos(x)

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y^{''}+6y^'+13y=30cos(x)

y^{^{''}} + 6 y^{^{'}} + 13 y = 30 cos (x)

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

Solution steps

y^{''} (x) + 6 y^{'} (x) + 13 y = 30 cos (x)

Solve linear ODE:

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

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

y^{^{''}} - 3 y^{^{'}} = 1

y^{^{''}} - 4 y^{^{'}} + 4 y = x

y^{^{''}} + 4 y = t^{2} + 2 e^{t}, y (0) = 0, y^{^{'}} (0) = 1

(d^{4} + 10 d^{2} + 9) y = sin (3 x) + cos (3 x)

y^{^{''}} - 10 y^{^{'}} + 25 y = 30 x + 12

What is the solution for y^{''}+6y^'+13y=30cos(x) ?
The solution for y^{''}+6y^'+13y=30cos(x) is y=e^{-3x}(c_{1}cos(2x)+c_{2}sin(2x))+sin(x)+2cos(x)