Question

Consider the differential equation y"+ 3y' + by = 0 where b is a real number. a) Find the value of b that makes the above differential equation critically damped. b) Solve the above differential equation for the value b=4 where y(0) = 1 and y'(0) = 1. Put the solution into the form Asin(ot+o).

Answer 1

steps given: y'+ 3y't by 20. characterstic equation is (s'+33 +b) yu) 20 > $+35+ b 20 comparing with staqrons twn zo 2 corab I 2&wn 23 wharo | {wn = 3/2 for critically damped 21 Step2. Step 2) {wn 23% 3 (1) Wn 2² Wn a32 But wna V6 2) 32 V6 b = 1

steps step2: Given y + 3y + by 20; y(0)=1 gel + 3y + 44 20 (2²+ 3D + H) 920 characterstic equation is D + 3D +420 m 43 m +420 (Dam) arli bo 3; Cid ma –3+ / 9-4(H) - ) 1 AC - 3 7 119-16 = 375 steps & Stepy. general solution is ylt) = 36+ ( cos(12+) + C sin Gzt) But 410) = 1 yo) e 360 (cicos (0)+c9 sinlos) 1 1 (4+0) step dylt) = -cje 3bt sint) 4h) - Gent cos/rift & Gezt cos ft) ²) But 460) 2) #C2 st sint" ylo- -, elo) -3), 4 eº los(0) + va cjeloste) 1 -39+Ger +0

1+ 3a = b 2 ) 5 Ca (But call - y) = 3€ eoskz t) + 5 sin @J+) 22 (vs cos (6 y + 5 sinks +) = 2*X5-656 | sin 156 +82-114) n ut) 20137 (sin (t +62-114) X