Question

1. Suppose that y= a (x-1)" is the power series solution of the following initial value problem. x-y+2y=0; y(t) = -2, y(1)=1 Find the value of az. 2. Suppose that y=0(x) is the solution of the following initial value problem. y" + xy - (sinx)y=0; y(0)=1, 7(0) = 3 Find the value of (0) 3. Let p be the radius of convergence for the Taylor series of the following rational function centered at the given pointxo Find the value of p. 4. Find a positive lower bound for the radius of convergence of a power series solution of the fol lowing homogeneous linear equation centered at the given point to (x+8)y"+(2+1)y + xy = 0, X0 = 1 5. Determine which equation is the general solution of the following Euler's equation. 2(x + 2)?y" - 2(x+2)y+3y = 0, where x +-2 (a) y = x+2p/2 cos ( Injx+2)+cz/x+2p/2 sin ( Infx+21) for any constants eande (b) y=c}\x+2/cos (In (x+21) +cz|x + 2 sin(\n [x+21) for any constants and ca (e) y =ox+2/1/cos (41 x+21)+clx+2/1/2 sin ( x+21) for any constants en und es (d) y = cos(31n x + 2) +csin (31nx +21) for any constants cande

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Answer 1

sol et y = Å an (x-1)" be a usole of xyl - y' + dyzo yll= -2, "'lla) neo et a-l=t then a=tti d. dy.dt, dy JO wo nco nog na 2 o in-2 n-i nant ta ant 50 > n(n-1) ant ท-1 nzo nel ท=4 so, we conclude that g= § ant se a soli of (&+1)y" ~ y' + 2y =0 @ w you mantos e gove E nima) dan put @ by ② & ③ we have +E nina) an nimuant Enant "it d& antao t"-1 + ² lantzo - E tren mant" + i les insanes # - Emwang p* Exent? 0 - (24 -1, + 2a) - 8 (ntra) - ( ) ) and (4) (n-2) 424 ) ? - 2 203 +880 - ,20 nag 12 nel D t" + [ n(n-1) ant" - nan nso nal nag - n=2 - 0 D nco nco no nal X nal 200 -0,=0 noil + (n+1) (0+2) antů + 2 ano # ag= amada #nzil - - dan - (n+1) (n-1) anti and O anta= no botell et has then az= -291 = -2 So, y al É ant"- Qot ait + Q3t+ - = Got ait + (91-90) 4 - 0 ++ -- y (4) = .(1-4 + --) +911+*******--) + y lala 2011- (x-)*+---) + 01 (6-1) + 31-12-414-18+--) As y10=-4 wo, 20 = - d d y' (1)= 1 = 0,= 1 + y (x1= -2+ &#'t--- + ++4**#'= -2+++*++++++ = 103= /3