# How to find the solution of the following differential equation? [closed]

DSolve[
{x''[t] + (k/2 - i(d + g(c + a Cos[t ω]))) x'[t] + i g a ω Sin[t ω] x[t] == 0},
x[t], t]


I tried with initial condition x[0] == x'[0] == 0 just to test it, but it's showing an error.

The solution is in terms of Fourier series with coefficients having Bessel functions of the first kind. $$Edit$$: Initially, its a first order differential equation.

x'[t]=-k/2(x[t]-x1)+i(d +g(c+ a Cos[t ω])) x[t],


where d, x1, c, g and a are constant. Since, the solution shows some Bessel function of first kind. So, I thought it requires a 2nd order differential equation to get that solution. But not sure if it is possible with first order derivative equation as well. so I differentiate the equation to get double derivative w.r.t. time as shown inside the DSolve. Its actual solution is written in a Fourier series: $$x(t)=e^{\text{i\phi } (t)} \sum _n x_n e^{i n \omega t}$$ where coefficient is $$x_{n}=\frac{x1}{2} \frac{J_{n}(-g a/\omega)}{i n \omega/k +{1/2} -i(g c+ d)/k}$$ and $$\phi = (g a/\omega)Sin(\omega t)$$.

## closed as off-topic by AccidentalFourierTransform, Lukas Lang, Mariusz Iwaniuk, Bob Hanlon, Henrik SchumacherNov 10 at 17:17

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – AccidentalFourierTransform, Lukas Lang, Mariusz Iwaniuk, Bob Hanlon, Henrik Schumacher
If this question can be reworded to fit the rules in the help center, please edit the question.

• Cos instead of cos and \[Omega] instead of \omega. – AccidentalFourierTransform Nov 10 at 16:46
• With the corrections made to the code, DSolve provides an answer. – bbgodfrey Nov 11 at 10:26
• @bbgodfrey Any suggestion to get the solution in Fourier series? – Off Topic Nov 13 at 16:48
• What boundary conditions do you wish? – bbgodfrey Nov 13 at 18:59
• @bbgodfrey, I have added more detail but not sure about the boundary conditions. – Off Topic Nov 14 at 10:23