Question

Consider the following initial value problem, (1 - 2)" + 3xy' - 8y = 0, 3(0) = 3, 7(0) = 0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals. (a) This differential equation has singular points at Note: You must use a semicolon here to separate your answers (b) Since there is no singular point at x = 0, you can find a normal power series solution for y() about <= 0, i.e., () = am " m - 0 As part of the solution process you must determine the recurrence relation for the coefficients am Enter your expression for am 1-2 am+2= 22 am (c) Use your recurrence relation to fill in the blanks below 02 = 95 = (d) Using your results above and the initial conditions, enter the first three non-zero terms of the power series solution for you)

Answer 1

So,? (x) = -8 solue by seis medron 1 - x) =" + 3^2 - Ýy = 0 -D ylo) = 3, y'lo) = 0 on companying to std. eyn play y" + qul y' + rial y =o on comparing to o blu) = (-x?) g(x) = 3x, (a) if xo is singulon kuined than Þ(x) = 0 > |- x = 0 = 1 = + ) (6) et sum of 6 is yle) = E am 2 - XC Am mo diff wi pi to z m -) y' (al = mam a 8W & SW ma) am 2 -3 a diff w. rodon mlm-1) aim zem? - É ma!

y" (x) = mm-1) an xm-? - 42 m2 given that y lol-3, yllo) = 0 then from @ yo) = @ 3 - Go + a, o + an o'ta.. =) ao = 3 then from ☺ y'll = 0 = 1a, + 2a, lol + 39 (0)t... => a = 0 . then from o - 1-x²) y + 3xy - 8y =o O using @ , 6 = (1-x²) Å m/m-il am xmi-2 + 3x e m am xm- me) mal => Se alm.) com xem-e - Ž. moms) om za + 3 ma

mm- m + 2 2 m (m2, me ) omes 2" - Z., man.) an za + 3 Žmom, zem - s . Om z" =0 => [204 + 60 X + B (ona) mana2) aman zam] - - (--) - - + = 4, + 3 ] - | ao + sa, t + 1 gam *1 = 0 2 + b c x to zm m=2 | O + (- 8am x ao + 8a, x mem - m2 => (202-840) + 1603 +39, -89,) x + E [lm+))(m+2) amiz - mim-1) am +3mam-8 cm2 m2 =o on comparing 29 -89,20 => a=49o. 603 +(34, -89,) = 0 => ag = şa,

(maillm+2) amaz-mim-1) +3m +8 ] am = 0 => amsa = m/m-1) -3m 48 am (mall/m+2) Om de ņ ņ 13 amtz m - um + 8 to ammo a (mto) (mm + 2) a but mzo => a = Dolor 8 ao = 0 but mai =) ag = blandt a, = Ža, put m=2 => au = 1/3 az slm ble m=3 => but -ts as = ħ a3 = a, (d) General son y(x) = is at a, x + ax² + ₂x²+ aux"+ Qx + -.. ao tan + 490 x²+ şax² + y oth + a, xv + a.. y

→ 4121 = ao [li + um? + 4 x2") + ---- ] am (*+ x + + ---) buh ao =3 , a,=0 3 + ( x) = 12 x^ + 4 x” + .... =>