Question

If `y=e^(mtan^-1x)`, check whether the equation `(1+x^2)y_(n+1)+(2nx-m)+n(n+1)y_(n-2) = 0` ?

Answer 1

The specified function does not satisfy the given recurrence relationship.

Explanation:

We have:

`y_0 = e^(mtan^-1x)`

And we wish to show that:

`(1+x^2)y_(n+1)+(2nx-m)+n(n+1)y_(n-2) = 0`

Where `y_n` represents the `n^(th)` derivative wrt `x`.

Let us test the base case `n=2` of the recurrence relationship, which gives us:

`(1+x^2)y_(3)+(4x-m)+6y_(0) = 0`

Now, let us test a specific case of `m=0 => y=1`, Then, all derivatives are zero, and the base case recurrence relationship is as follows:L

`(1+x^2)y_(3)+(4x-m)+6y_(0) = (1+x^2)0+(4x-0)+6 = 0`
`" " = 4x+6`
`" " cancel(-=) 0`

Hence, the specified function does not satisfy the given recurrence relationship.