Question

Question 5 8 pts Let y be the solution of the equation y'' – 4y + 3y = 0 satisfying the conditions y (0) = 1 andy (0) = 3. Let f (x) = e->y (x). Find the value of the function f at r = 1. 8 pts Question 6 Let y be the solution of the equation y' + 4y = 0 satisfying the conditions y (0) = 0 and y (0) = 2. Find the value of the function y at x =

Answer 1

♡os Gluen DE 4"-44' +34 20 then by D- Opencetor p²y-4D4+3y=0 (0²_40 1D + 3)4 20 egn is. It's awsillen m²_4m +3 o M-3)(m-1) -0. :: Roots Roots are m=3 & ma= 1 . E. solution is 412) = ciex then 332 Y'2x) =Bcie + czere Given 410) - 1 8 yco) - 3 4107-1 = geoteze Cl +(2=1 4209-3 = 3ue teze 34+ C2 = 3 2 Solving ① & ②

24 = 2 CI-1 e Ico (2=0 320 4(x) = 4e tlere e 3x to 는 32 14(x) z fuz) - 3 - e 42x) five) zēs e 32 at x-1 FLI)=e ēs es fu) - walues of F at n=1 is

224 Given DE 15 4"'+4420 ie (²+4) y=0 Auxillary eqn is m?+4 70 m = 42 Roet's are complex pair is, .: Sol in 4(x) = CI LOS 2x (2 sinax ycz) = -20, sinax +262 cos2x. Given 410) -0 R 4'60)=2 416) 20 - ci Cos (0) + Casino) Ci (1) to 4120 84202 = 2 2-2c, sin (0) +212 Coslo) = 0 +21260) 2c2= 2 C2 - 1 Y(x) = sinar

at xi 11/4 4(1/4) = sin(exy 4 esin (1/2) 4(1/4)=1 value of y at 7-14 is 1