Differential eqtn (What's wrong with my calculation?)

[athrun200]

I would like to ask why I obtain a different answer from the book?
If I am wrong, how to obtain the correct answer?

Homework Statement

Homework Equations

The Attempt at a Solution

 

[vela]

Rewrite the differential equation slightly to get
$$\rho^2u''(\rho)-l(l+1)u(\rho) = 0$$
Then substitute a trial solution of the form $\rho^r$ and then solve for r.

 

[athrun200]

Rewrite the differential equation slightly to get
$$\rho^2u''(\rho)-l(l+1)u(\rho) = 0$$
Then substitute a trial solution of the form $\rho^r$ and then solve for r.

Is it another way to solve ODE?
What is the name of this method?

I can only remember to sub e^r:tongue2:

 

[vela]

Euler equations

http://tutorial.math.lamar.edu/Classes/DE/EulerEquations.aspx

 

[HallsofIvy]

You basic error is trying to apply a method that works for equations with constant coeficients to an equation with variable coefficients. As a result your characteristic equation, which should not involve the idependent variable, is wrong and everything after that is wrong.

 

[athrun200]

You basic error is trying to apply a method that works for equations with constant coeficients to an equation with variable coefficients. As a result your characteristic equation, which should not involve the idependent variable, is wrong and everything after that is wrong.

how about this one?
How to solve it?

 

[Gib Z]

There is no simple answer for that one, though you can express it in terms of http://mathworld.wolfram.com/ParabolicCylinderFunction.html .