Differential equations - verification of homogenous solution

vid

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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.
 

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tkhunny

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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.
"Verification" is a matter of calculating the derivatives and substituting.
 
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vid

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"Verification" is a matter of calculating the derivatives and substituting.
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

. . . . .\(\displaystyle y_n^{''}\, +\, \big(2\, \coth(x)\big)\, y_n^{'}\, =\, 0\)

and

. . . . .\(\displaystyle y_n\, =\, \left(\dfrac{1}{\sinh(x)}\, \dfrac{d}{dx}\right)\, \left(e^x\, +\, e^{-x}\right)\)

Then y_n has a form of \(\displaystyle y_n\, =\, -2\, \coth\left(x^2\right)\). Then I calculated \(\displaystyle y_n^{'}\) and \(\displaystyle y_n^{''}\) and I inserted it into differential equation. The result which I obtained was \(\displaystyle -4\, \mbox{csch}(x)^4\) which is not equal to 0 (DE doesnt have homogenous solution). Is it correct logic?
 

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tkhunny

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I don't understand your need for values for these parameters. Why not just use them as they are? You have not done the exercise if you first simplify it and then do the demonstration. You must do the demonstration for ALL values of the parameters.
 
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