Question

1. (a) Determine the largest x-interval where the initial value problem has a unique solution: 1 1 (22 – 40) (6) + y(5) + (x + 1)y" + e*y' + (tan x)y In (x – 1) x2 9 = = = with y(2.5) A, Y' (2.5) B, y" (2.5) C, y'" (2.5) D, y(4) (2.5) y(5)(2.5) = F, where A, B, C, D, E, and F are some known constants. E, (b) Determine whether the set of functions {5, sin’ x, cos 2x} could form a fundamental set of solution of a linear differential equation.

Answer 1

solution 1 (6) Let 9.5+2 sin rete cos 220 year o 250, tosinte tą cosas singx=9Bimecoco y'a 2 -243 differentiating equation 06.3 to re we get y=0+&xa sin cosul - 29 singue y = $casinue cosxe) – 26 sinar Ca sinan - 2 czsinare {: sinee- ' = (62-263) sin23c y's A sinan put A=C2- {"=84 60829 9": Y"=-4.7" { y'= a sinon y" +9y'=2 required differential equation =-4 A sinare this is

Solution 110), Griven differential equation is L 405) + (x+1) oyutery +{tanzely = x = 9 I (re?-40) 4147 16) 7 ln (2-1) it can be written as (set) y 2(x-4) H y" + en + dana, y = 22-3X**3) *(x-4) K(x-+-(4-1) with yout 1 yet companing equation o yº't ganyos) tapy?" +ą, ew y"42,(wy "taway'tgary we have ㅗ as(x) H(x-4) In (x-1) 92(x)=0 +1) 94(H)=0 93(a) = 2(4-4) % % (x) = tem eu win 4) 24x-4) Ke ,air) to Rox) = lo (2-3)(2+3) aşcu) wis discontinuous at point xe (-401) U{2,43 9€ 40,43 HE {0.43 & Ron) $0 Largest interveel az cx) is dis continous out joint us des continuous at point qocal ao (n) is des continuous at point ac qo es is dis continous at point ne {~3,33 wslere given differential equation has solution is I= (1,8) U (2,3) 0 (3,4)U(4,00) eenue largest interval where given differential equation was unique the solution is [(4,0) Al