Question

8.6.9 Question Help Classify each singular point (real or complex) of the given equation as regular or irregular. (x2+x-6) 2y"' +(8x + 24)y' - 6x2y = 0 Identify all the regular singular points. Select the correct choice below and fill in any answers boxes within your choice. O A. x= (Use a comma to separate answers as needed.) O B. There are no regular singular points.

YOU MUST SOLVE ALL PARTS OF QUESTION!
Part A) Find all regular singular points
Part B) Find all irregular singular points

DIFFERENTIAL EQUATIONS / Linear Algebra

Only people that are proficient in DIFFERENTIAL EQUATIONS should even attempt to solve. No beginners or amateurs allowed.

Please write clearly and legibly. No sloppy Handwriting. I must be able to clearly and easily read your solution and answer.

Circle final answer.

Answer 1

Solution :- Given differential equation is :- (x2+x-6) ²4" + (8x tau lyl-6x²y = 0 =) Y" + (84+24) (x24x62 ý 6x2 (t+x2 Combering with y" +pedy' + Q(x) =0, we gets Plana 84124 QGx) = -642 (se?+x-6)2 (x²+x-6) 2 P(V) 8 (+3) Q(1) - 6x2 (x+3)*(x-2)8 (X+3) (7-2) 2 limit of P(x) and Q(x) does not exist attr- x. 2-3 x=81-3 are ringular points Now, check at x 2 lim (x-2) P(x) = 8 (1+3) (1-2) m 2 (0+3)2(0-2012 lim 8 ang (x+3)(x-2) does not enest x=2 is irregular singular point

Result: and let no be the singular point of g + pox) y't Q (120 lim (x-koll(a) er lim (x-16)² Q cx does not exist 14 Xe x no then ur do is the insegular singular point] Now, check at x= -3. clim (x+3) P(x) = lim 8(x+3)2 44-3 (+37244212 lim 8 xt-3 (x-212 N-3 8 (-5) 8 25 Cerust firmikely) clim (x+3) ? Q(x) = lim = (x+372 6x2 (1+372122)2 H1-3 14-3 lim - 6x2 at-3 (0-2)2 -6x9 -54 25 (exist fimilely) (-5) =) X=-3 j the regular Lingular point- Result ir let to be the legender Singular boird of y "+pca) y't&w=0, lim (X-86) P (A) and lim (x-76)²2 (2) both and *1 Ko are x720 exist finitely then xato is the regular singular boind of giren diffemtial eenetrend. - A - 2 is teme
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