Question

(1 point) In this problem you will solve the differential equation or @() (1) Since P(a) 0 are not analytic at and 2() is a singular point of the differential equation. Using Frobenius' Theorem, we must check that are both analytic a # 0. Since #P 2 and #2e(z) are analytic a # 0-0 is a regular singular point for the differential equation 28x2y® + 22,23, + 4y 0 From the result ol Frobenius Theorem, we may assume that 2822y" + 2223, + 4,-0has a solution of the form y zr Σ。eak which converges for-e 0 R where r and R are constants that will be determined later r Σ。。о 얘 z "mo 2822y" +2.2y, + 4,-Q we get that (2) Substituting y # The subscripts on the c's should be increasing and numbers or in terms of n. (3) In this step, we will use the equation above to find the indicial roots and the recurrence relation of the differential equation. (a) From the equation above, we know that the indicial roots of the differential equation are (in increasing order)- and r = (b) From the series above, we find that the recurrence relation is for (4) The general soltion to 2822y2 4y Ois and converges on the interval Use -inf foro and inf for co

Answer 1

28χ2 Paga- I y 2. 1나 aore not analytic at a% Point Dissesrentiata two times coith oraspact to e get titute the value os , y' y Sebet

K-o age-2 2 cohich is iantity Equating to 3ho the cesSicient os lovest degree tam 91 +4] = o

2. 914(1+2)-1 (1+2) +23 14 (91+2)-(2)+23 14(9+3)2-14 (S7+3)+ 2 91 9 (-14)+2 2c

the auiredl Soluth on is 23.2646(23.2646) (745046) 2 っc -(O33) (1.83(2.83 (23.2646) C74.50Y6)(iS376) 4.7846 (4.786) (37.5446) 一@i7) (1.17)(2/7) (4.7846) (37. 5446) (38.30y6) 3 Page-y