Question

Hi everyone! xy"+(sinx)y=0
1)how did they know the sin/x can be written as a power series? Is there an easy way to tell?
2)and how do they know the ordinary point is at x=0 ? Thanks! :o)

Answer 1

@rational @kainui @zepdrix @amistre64
In the general form y"+P(x)y'+Q(x)y=0
A point Xsubzero is called "ordinary" if both the function P(x) and (Q(x) can be written as a power series with a positive radius of convergence and the point Xsubzero is otherwise called a "singular point" if both P(x) and Q(x) CANNOT be written as a power series.

The examples they give are xy"+(sinx)y=0 where after dividing by the leading coefficient, P(x)=sinx/x and since sinx/x can be written as a power series, there is an "ordinary point" at x=0

so, 2 questions...
1)how did they know the sin/x can be written as a power series? Is there an easy way to tell?
2)and how do they know the ordinary point is at x=0 ?

Answer 2

hi rational!
are my questions hard to answer?

Answer 3

yes haha
@Zarkon @eliassaab

Answer 4

well the reason I am asking this is because of this...
if you get a question that asks you where the singular or ordinary points are for the DE
(x^2-4)y" + xy' - y = 0
you have to be able to both figure out if they are actually ordinary points or singular point by knowing if x/(x^2-4) can be written as a power series, and then also find where the points are...I was trying to sum up the problem to make it easier

Answer 5

grrr

Answer 6

hello :>

Answer 7

sin(x) has a well known power series definition, and u divide everything by x, so u are shifting the degree down

Answer 8

you aren't making fun of me this time dan?