Question

"Which of the following statements correctly describes the behaviour of the differential equation near the singular point [. . .]?"

http://www.tiikoni.com/tis/view/?id=33f6d61

I understand how to get x_1 and x_2, but could someone please tell me why the multiple-choice parts are C and B, respectively?

Answer 1

well it's good that you know where \(x_1=-2,x_2=2\) come from -- this is due to the fact the coefficients of \(y'',y'\) vanish at these points

Answer 2

By vanish at those points you mean have undefined behaviour at those points, right?

What about the multiple-choice parts, though?

Answer 3

vanish i think means go to zero

Answer 4

well in general when we say vanish it means go to nothing (zero)

Answer 5

for example, if y->0 as x-> 2 , y vanishes at x = 2

Answer 6

im not sure what you mean by 'present'

Answer 7

present?

Answer 8

for \(x_1=-2\) let \(\displaystyle y=(x+2)^r\sum_{n=0}^\infty a_n(x+2)^n=\sum_{n=0}^\infty a_n(x+2)^{n+r}\) so \(\displaystyle y'=\sum_{n=0}^\infty (n+r)a_n(x+2)^{n+r-1}\) and thus \(y''=\sum_{n=0}^\infty (n+r)(n+r-1)a_n(x+2)^{n+r-2}\)

Answer 9

@oldrin.bataku, sorry for the late response. It seems like what you showed is the algebra on which the multiple-choice questions/answers are based off of, but I still haven't fully made the connection. Could you tell me what the logic is for knowing which answer to choose, based on the algebraic manipulation you just showed?