Question

Exercise 7.3.101: In the following equations classify the point x = 0 as ordinary, regular singular, or singular but not regular singular. a) y" + y = 0 b) x3y" + (1 + x)y = 0 c) xy" + x y' + y = 0 d) sin(x)y" - y = 0 e) cos(x)y” – sin(x)y = 0

Answer 1

as x ko, Consider a second onder ondinany differential equation y" +PGx.y' +008).y = 0 If P(a) and Q(a) nemain finite at x=xo, then xo is called an ondinary point. If either Play or Q.c«divenges then do is called singular point. If eithen P(x) or Q(x) divenges as but (x-xo) P(x) and (x-xo) 8(x) remain finite as xxo, then x= xo is called a regular singular point. 2-

( @y" + y'= 0. P(x)=0, 8(x) = 1 At x=0, P(x) = P(o)=0 Atx=0, 8(x) = 8(o)=1 is an ordinany point b xpy" + (1+x). Y = 0 →y" + o. y' + 1**.y = 0 Hence, 2 = 0 23 lim 1ta lim 23 23 xo 2-0 24->0 P(x) = 0, Q(x) = 1+x lim Q(x) = *) Hence, lim 8(x) does not exist x=0 is a singular point. Now, lim (x-o) P(x) = din (4-0)*Q(2) = lima? (**) lim Therefore, lim x.o=0 2 --> 0 2 --> 0 1+x xto 21->o t lim no 21->0 Therefore, Hence, lim (x-o) Q(x) does not exist. singular but not regular singular,

(C xy" + x 4 ty y" + xy + 1 h y =o. P(a)= a 4, (*) - Atx=0, P(1) - P(0) - 0940 lim lim o(a) 2 -> xo Hence, lim 8ca) does not exist Therefore, is a singular pt. Now, lim x-6) P (n) = lim x. P (2) lim x. x4 Ho nob >0 xo lim (x-o) Q(x) = lim x². 12 lim no Therefore, 220, is a regular singular point. (d) sin(a).y"_y = 0 > " + 0.4' + (-sin(x) = P (a)=0, Q(x) =- Ata=0, P(x)=P(o)= sin(a) lim Q(x) = lim Esinca Xo 2 -> lim ato (- does not sin (2) Ho since, lim Q(x) = exist, therefore, a=0 is a Inns) singular Jan point.

lim x. P(x) lim x.o=0 xo Xo sinx lim xr. Q(x) no Xo tim » [armas ) - hieman -) ( (lim 'n so kina 12 so 1 kina 270 1 1 -0.1. Hence, xo is a regular singular point

sin(a) Cos (2) tan (n). (e) cos(a), y" -{sin (x)} y =o. » Y+0. y' + [ - Bin(x)].y=0. P(x)=0, Q(x) = - Atx=0, P(x)=P(o=o. Si At x=0 Qin) = 8(0)=- tan (o)=0 Hence, x=0 is a singular point. dim. Geno) P(x) lim a. P(a) lim x.o 2-) Ho

xo him (2-0)8(*) lim x². (-tan (x)). lim (tanx). x² lim tani(a) Elim 3 > 0 Ho az xo Thus, lim (1-0) 8(x)=o. 20 is a a regular singular Hence, x=0 point.