MATH2100 - Applied Mathematical Analysis Final Examination, Semester 2, 2014 1. (a) (16 marks) Sketch the trajectories of the following system in the phase plane, indicating the direction of flow, and classify the type and stability of the critical point at the origin. (%) = ( 3 2 ) (3) State the equation of any straight line trajectories and find the equation of the straight lines where the tangent to the trajectories is horizontal or vertical. Find those points in the phase plane for which 3y1 - 42 is increasing. y = Ay where A = 1+ 2'). E-values: |A-XII 2 ). Here = |4- > 1 = ( A - 2 ) (1 - 4) - 3 = >2-64+5=(4-5)(X-1) 3 2-> 7 = 1, 5 are e-values - both real and positive .: Crit. pt. at origin is an improper node which is unstable. A=0 OR: P= G A = 6 > 0 q = det A = 5 > 0 D = p2-4q = 36-20=16>0 Crit. pt at origin is a repulsive node which is unstable E-vectors : > = 1: 0 = (A-I) x = - Set u = 1 => { "' > = (-3 ) >=5:0= (A-5 I)x = (3 3 1 1 ADO Node ( R) P )(v) => V + 3 4 = 0 is e-vector. = > v = 4 3 -3 ) ( y ) : set u= = > x(2) = ) 5 e-vector. Gen. Solt: y = { x ") e+ + > x (2) est St. live trajectaries: When B= obtains st. I mie trajectory y == 3y 1 corresp to e-vector x") - arrows away from origine. Page 2 of 18
MATH2100 - Applied Mathematical Analysis Final Examination, Semester 2, 2014 (Working Space Only) when I = 0 obtain st- line trajectory corresp. to orgenvector x(2) - arrows away from y, = y 52 oriqui. 1 As t ->00 term x (2) dominates- so all trajectories /1 to st line y, my as It -> 00. As (-) - to term "1" dominates to all trajectoires' are tangent to st line y =- 3y, at origin. 2 Finally from chan's rule slope to trajectories is given by = . 92 dyr dy y = 3 g + 2 = = 0, y= = = 3y, 1 1 4y, +12 o, y2 == 4y , -: Tangent to trajectaries is horizontal along st. live y = - 3/2 y, and is vertical along st. line giving phare diag. shown: 42 =- wy, A 7 y2 = y, 2 y, 1 1 6 y2= = 3y, If s = 39 , - 4 y _ we have s = 39, - 4/2 = 3(49,+12)-4 (3), +2/2) = 12y, +39-12y,-8y2 == 5y2 => s is increasing when y < D - Le s b Ty2. increasing at all pts halb 2. Lower Page 3 of 18 of phase-plane. y
MATH2100 - Applied Mathematical Analysis Final Examination, Semester 2, 2014 1.