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To calculate derivatives I assumed that n=1, A=1, B =1, But I am not sure if I can make assumption for A,B. (I calculated derviatives without this assumption and the results of the derivatives were very complex). After assumption n=1, A=1, B =1 my differential equation has form . . . ...

I would like to ask how can I verify that the differential equation: . . . . .y_n^{''}\, +\, \big(2n\, \coth(x)\big)\, y_n^{'}\, +\, \left(n^2\, -\, 1\right)\, y_n\, =\, 0 has the homogeneous solution: . . . . .y_n\, =\, \left(\dfrac{1}{\sinh(x)}\, \dfrac{d}{dx}\right)^n\, \left(Ae^x\, +\...