Question

2. (3+4+4+4 pts) In this problem, we discuss a method of solving SOL equations known as Reduction of Order. Given an equation y" +p(a)y' +9(2)y = 0, and assuming yi is a solution, Reduction of Order asks: does there exist a second, linearly-independent solution y2 of the form y2 = u(x)41 for some function u(x)? See Section 3.2, Exercise 36 for reference). We'll now use this to solve the following problem. (a) Consider the SOL differential equation sin(x)y" — 2 cos(x)y' – sin(x)y = 0, 0 < x <a i. Show that y1 = cos(2) is a solution. ii. Suppose that y2 = u(x) cos(x) is also a solution. Find the re- ducible second order equation that u(x) must satisfy by plugging y2 into the given SOL equation. iii. Solve the equation you found in (b) to find a suitable u(x) and thus a suitable y2. Show via the Wronskian that yi and Y2 are linearly independent. (b) More generally, if yı is a solution to y" + p(x)y' + q(x)y = 0, show that if y2 = u(x)41 is a solution, then u(x) satisfies the reducible second order equation yil" (x) + (y1 + 2yt)u'(x) = 0

Answer 1

- Geven that Que Yomia com as y, lm =- stam a f (g) = - Coom Now - (Simny - (2 loody - Binn) yao; ocnen - - if y, n) a Coon is a solution of ①, they (Sinny" - P los ) y, sinnly, and for genet - Now Sinny = sinul-Coon) - asian loon - 6 -(2 conly,' = -12 Coon) I-Sinn) a Sinn Cosm - - 1 sinny, a - Sinnloon - inny, - 2 losny, - Sima) y = - Sinaloon A2 Sinn Coon-sinn log on Yomi coon is a solu Hom of in differential equation, Ya mea. Mm) loon ylna - km sinn & M'lm) Coon "m) a (Coon) M" In ) - (Sinn Malm) - (Sinn) mallos rum) - Coon) m n 12 sinn Mlm) - Koon) HIM) is Yg lm) in a solution of ①, then

(in)y" - 2 Cor) y,' - 1 sinn)y, no & linne Consulmi - Weinn a'w) - (Common), od comm ( M'w) toon - (SM) MIn) - sinn Immo ton) o (Sinn Conn) 41m) - 2 Ming no Sinan Min) - u Mlm no which is the reduble second order equation. (1) let vimo. Ming; then ® reduna to ( Sinan) vlny - Yvirlaun Sinda). vllus. Uning n drin u dha u loped an am In luni 2 ln Cope, dnt Lot del t e 1 nap a C ( Copee din of cost an) 2 T + Conan Haam Copla vinde il suo magis , tone G (sola-1) eum (, Ser'-) Mini - Iltamnin)

E Yalny a G (sinn-n cosa) + lg leon Com с ( 4 см іn и Co . Co. , ися 4 с м - су у tо) и с har team оси < 17 Y the and I are linearly independent solution given differential equation,

+- ( - Kota ng anh ngt ihn hier ('hribtihadt hat i we geti Using all there in Ye lm, 2 My" 40"y, ang PART D osth lun b hould tch og sepanjen u hwn a ruled one o os hombo hrudd to h 79 Meer onth puf unin (9)

i From i we have 4, ") + 1 P 38 oy,') u'=0 cohich is the required reducible second order equation TTTT