Showing that a bessel function satisfies a particular equation

[1Kris]

Hi, I'm stuck on this question from a calculus book;
Show that y'' + ((1+2n)/x)y' + y = 0 is satisfied by x^-nJ_n(x)

Is it correct that when I differentiate that, I get these:
y= x^-nJ_n(x)
y'=-x^-nJ_n+1(x)
y''=nx^-n-1J_n+1(x) -
x^-n(dJ_n+1(x)/dx)?

The Attempt at a Solution

Equation in question becomes:
x^-n(2(n/x)J_n+1 - J_n - ((1+2n)/x)J_n+1 + J_n)

= x^-n(-x^-1J_n+1)
but this isn't 0.

Sorry if I'm repeating myself here but I could just do with some kind of a pointer.
Thanks, Kris

 

[e(ho0n3]

Shouldn't y' = -nx^-n-1J_n(x) + x^-nJ'_n(x) ? You have to use the product rule.

 

[vela]

I think the OP is using some property of the Bessel functions to get that expression for y'. I could very well be wrong though; I don't remember much about the Bessel functions.

 

[vela]

Just crank it out, without resorting to any special properties and relations of the Bessel functions. Use the product rule to get y' and y'', plug them into the differential equation, and simplify.