What did I do wrong in this Bessel equation?

[yungman]

I need to convert$$x^{2}y''+2xy'+[kx^{2}-n(n+1)]y=0$$ using $$y=x^{-\frac{1}{2}}w$$ to a normal Modified Bessel Equation and I cannot get to that. I check many times and I must be having a blind spot!

This is my work:

$$y=x^{-\frac{1}{2}} w \Rightarrow y'=-\frac{1}{2}x^{-\frac{3}{2}}w+x^{-\frac{1}{2}}w' \Rightarrow y''=\frac{3}{4}x^{-\frac{5}{2}}w-\frac{1}{2}x^{-\frac{3}{2}}w'+x^{-\frac{1}{2}}w''-\frac{1}{2}x^{-\frac{3}{2}}w'$$

$$x^{2}y''+2xy'+(kx^{2}-n(n+1))y=0 \Rightarrow$$

$$[\frac{3}{4}x^{-\frac{1}{2}}w-\frac{1}{2}x^{\frac{1}{2}}w'+x^{\frac{3}{2}}w''-\frac{1}{2}x^{\frac{1}{2}}w']+[2x^{\frac{1}{2}}w'-x^{-\frac{1}{2}}w]+[kx^{\frac{3}{2}}w-n(n+1)x^{-\frac{1}{2}})w]=0$$


Grouping w'', w' and w terms:

$$x^{\frac{3}{2}}w''+(-x^{\frac{1}{2}}+2x^{\frac{1}{2}})w'+(\frac{3}{4}x^{-\frac{1}{2}}-x^{-\frac{1}{2}}+kx^{\frac{3}{2}}-n(n+1)x^{-\frac{1}{2}})w=0$$

Multiply by $$x^{\frac{1}{2}}$$ This give:

$$x^{2}w''+xw'+(\frac{3}{4}-1+kx^{2}-n(n+1))w=0$$

I can't get the standard Modified Bessel equation because the 3/4-1 in the w term cannot be cancel out.
Please tell me what I did wrong.

Thanks
Alan

 

[ystael]

Your arithmetic is correct (except that in the first line you have an exponent $$-\frac23$$ that should be $$-\frac32$$, but later steps are correct).

Is there any reason why the $$n$$ (order) in the equation you end up with is required to be the same you began with? I notice that your problematic constant term is $$-n^2 - n - \frac14$$, which factors nicely.

 

[yungman]

Your arithmetic is correct (except that in the first line you have an exponent $$-\frac23$$ that should be $$-\frac32$$, but later steps are correct).

Is there any reason why the $$n$$ (order) in the equation you end up with is required to be the same you began with? I notice that your problematic constant term is $$-n^2 - n - \frac14$$, which factors nicely.

You are right, I was blind! The final equation is with $$(n+\frac{1}{2})^{2}$$!

It must be old age! I read the final equation wrong and I just kept concentrate on what I did and never look at the final equation! Wasted almost an hour on this! The only excuse I can come up is I am 56! Too old! BUT I am not old enough to conceed to you guys yet!

Thanks a million.

Alan