I would like to ask how can I verify that the differential equation:
. . . . .\(\displaystyle y_n^{''}\, +\, \big(2n\, \coth(x)\big)\, y_n^{'}\, +\, \left(n^2\, \, 1\right)\, y_n\, =\, 0\)
has the homogeneous solution:
. . . . .\(\displaystyle y_n\, =\, \left(\dfrac{1}{\sinh(x)}\, \dfrac{d}{dx}\right)^n\, \left(Ae^x\, +\, Be^{x}\right)\)
for degree \(\displaystyle n\, \in\, \mathbb{N}.\)
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 don´t really understand how it can be verified. I will be grateful for all help.
. . . . .\(\displaystyle y_n^{''}\, +\, \big(2n\, \coth(x)\big)\, y_n^{'}\, +\, \left(n^2\, \, 1\right)\, y_n\, =\, 0\)
has the homogeneous solution:
. . . . .\(\displaystyle y_n\, =\, \left(\dfrac{1}{\sinh(x)}\, \dfrac{d}{dx}\right)^n\, \left(Ae^x\, +\, Be^{x}\right)\)
for degree \(\displaystyle n\, \in\, \mathbb{N}.\)
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 don´t really understand how it can be verified. I will be grateful for all help.
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