(Sturm-Liouville) Problem posted below. My first inclination is to divide through by x to put it into the form of a Cauchy-Euler equation, but I don't know if I'm allowed to do that (that probably sounds silly, oh well).

Question

mendicant_bias · Answer

$$x^5y''+5x^4y'+\lambda x^3y=0$$

mendicant_bias · Answer

(Again, first inclination is to divide by x^3 through to put it into the form of a cauchy-euler problem, is this allowed/doable?)

mendicant_bias · Answer

@dan815

dan815 · Answer

ya

mendicant_bias · Answer

Alright, cool. So then$$x^2y''+5xy'+\lambda y=0$$

dan815 · Answer

ya this shud have a simple soln

mendicant_bias · Answer

Yeah, just haven't done cauchy-euler in a while and am trying to remember the general format of it/how it works

dan815 · Answer

y=X^n

mendicant_bias · Answer

$$y=x^m$$

mendicant_bias · Answer

Yeah, I gotchu, I think I remember

mendicant_bias · Answer

$$m(m-1)x^2x^{(m-2)}+5mxy^{(m-1)}+\lambda x^m=0$$

mendicant_bias · Answer

Simplifying through: $$(m^2-m)x^m+5mx^m+\lambda x^m=0$$That doesn't seem right, one moment.

mendicant_bias · Answer

Oh nevermind, divide across by x^m, I think? What's the justification for that, can it be shown by bounds or otherwise that x^m is not zero, or is that part of the general conditions on the problem that need to be fulfilled for this to work?

dan815 · Answer

x^m is not zerno unless m is -infinity so

dan815 · Answer

so u can just say X^m is never zero

dan815 · Answer

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