Question

Problem 3: Find a solution to the IVP day d.x2 + dy dc + y = 0, y(0) = y'(0) = 1.

Answer 1

dzy Given - tdy dn y (0)=y'co)=1 2 ty=0 dx Solh To find :- IUP Sælution Let's find solution of the associated homogeneous Squation yTy't y=0 The Auxilliagy equation es m² + m +1=0 reats are - b ± √b²ual 2 a a=1, b=1, C=1 -1 +7-4 m ز - ۔ 2 2 m -1 Ź Bi 2 Treets are 1-3 i and 1731 2 2 2 The solution is then m,= -1-137, m = -1+531 complemt ary e thel CicOsY2x+625 inz real part is -1/2 & Imaginary el from the intial condition y col=1 Seltje ase 2 y (o)=1 J 9

Sub Sliding 010) and you ylt=ėth (1,628 [zł +63 20024) ، در بر as e° / с, Les o t sino) y (0) = 1 1CC) 19 = 1 hol get Now differentiating Eq -t/2 g'lt) = d be C, costat t t ) 2 dt d Bin St Le-th C₂ 2 dt _th a leth =cd dt sind dtice t/2 sinis 3 th é 1 t e زاده) =Cl 2 2 t/2 Les Bt t/2 sin Bt Le 2 2 2 os 3 t) +ę y to [12 + C2 [ B e y! 10) = Ci [ - - - I e sino 1 = Ci[0-%] +9/21 - 2 Luz e sino - Léº caso] tc e caso [ I e 2 2 با -16, +53 C2 17 2 2 3 2 = -C, -C, + C2 = 2+1 3 3 L=13 13

Substituting cand o in quales y get 1 102 3 t t sinly con Dusion y Lt) = 0 ( cas lit + Band {PLEASE UPVOTED