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For what it's worth, using the same substitution as Chip Hurst in his Apr 22 answer, and after some coaxing, I got to this solution: solx = Function[x, 1/(4 Sqrt[ m^2]) (2 C (1 + 2 Sqrt[m^2] Cos[x]) Cot[x/2]^(-2 Sqrt[m^2]) - ( 2 C (Sqrt[m^2] - 2 m^2 Cos[x]) Cot[x/2]^(2 Sqrt[m^2]))/(-1 + 4 m^2) + ((m^2)^( 3/2) (1 + 2 m Cos[x] + ...
I think I have solved this ODE (I didn't verify the solution). The problem with DSolve is Integrate was not terminating for this inhomogeneous equation. So what I did was solve the homogeneous equation, then applied variation of parameters described here: homode = -16c*m^2Cos[x]k[x] - c(Sin[3x] - 7Sin[x])k'[x] + 4c*Cos[x]Sin[x]^2k''[x] == 0; homsol = ...
1. See the function ComplexExpand ComplexExpand[Re[x + I y]] gives x 2. Also you can try Refine combined with Element: Refine[Re[x + I y], Element[x, Reals]] (*x - Im[y]*)
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