I would like to ask how can I verify that the differential

equation:

. . . . .[tex]y_n^{''}\, +\, \big(2n\, \coth(x)\big)\, y_n^{'}\, +\, \left(n^2\, -\, 1\right)\, y_n\, =\, 0[/tex]

has the homogeneous solution:

. . . . .[tex]y_n\, =\, \left(\dfrac{1}{\sinh(x)}\, \dfrac{d}{dx}\right)^n\, \left(Ae^x\, +\, Be^{-x}\right)[/tex]

for degree [tex]n\, \in\, \mathbb{N}.[/tex]

How to start this problem? Should I calulate first and second derivative from d/dn * y_n and later try to insert into equation? I donīt really understand how it can be verified. I will be grateful for all help.

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