Finding general solution of an ode using substitution

[honeypuffy]

Homework Statement

By making the transformation u= x^αy where α is a constant to be found, find the general solution of[/B]

y'' + (2/x)y' + 9y=0

The Attempt at a Solution

I've worked out y,y',y'' and subbed them into get

x^-au'' + x^a-1(2-2a)u' + x^-a-2(x^2-a(a-1))u =0

but I don't know what to do from here.

Any help would be greatly appreciated.

 

[Orodruin]

Can you perhaps chose $a$ in some intelligent fashion to simplify your equation?

 

[honeypuffy]

yes sorry I forgot that bit I had worked out that a should be -1 and there's a typo it should be a(a+1)
which would simplify it down to

x^1 U'' + x^1U=0

then I had the thought to divide by x to get U''+U=0

but I got stuck here because I wasn't sure on what to do. Should I change back to y?

 

[Orodruin]

No, you should solve the differential equation you obtained for u. Reinsering y would just give back the old differential equation.

Also, your original equation had a 9. Where did that go?

 

[honeypuffy]

OK so I would have U''+9U=0

with general solution of U=Ae^3x+Be^-3x

and U=y/x so y=(Ae^3x+be^-3x)/x

 

[Orodruin]

No, this does not solve the differential equation.

 

[honeypuffy]

sorry U should be Acos3x+Bcos3x

 

[Orodruin]

Correct. For equations of this type in general, you may want to have a look at spherical Bessel functions.