Semester One Final Examinations, 2012 MATH2010 Analysis of Ordinary Differential Equations THE UNIVERSITY OF QUEENSLAND AUSTRALIA This exam paper must not be removed from the venue Venue Seat Number Student Number Family Name First Name School of Mathematics & Physics EXAMINATION Semester One Final Examinations, 2012 MATH2010 Analysis of Ordinary Differential Equations This paper is for St Lucia Campus students. Examination Duration: 60 minutes Reading Time: 10 minutes Exam Conditions: For Examiner Use Only Question Mark This is a Central Examination This is a Closed Book Examination - specified materials permitted During perusal - writing is not permitted at all This examination paper will be released to the Library Materials Permitted In The Exam Venue: (No electronic alds are permitted e.g. laptops, phones) An unmarked Bilingual dictionary is permitted Calculators - No calculators permitted Materials To Be Supplied To Students: none Instructions To Students: Answer all the questions. Answer all questions in the space provided. Use the back of pages it space is insufficient and/or for rough working. Credit will only be given for work written on this examination script. All questions carry the number of marks indicated. This exam is worth 64% of the assessment for the course. Page 1 of 9 Total
MATH2010 Analysis of Ordinary Differential Equations emester One Final Examination, 2012 (a) (16 marks) Setting 3/1 = V, 1/2 = 9, express the second order ODE ÿ - 3? + 2y = 0 as a system of DEs y = Ay, y= (*) for a suitable 2 x 2 matrix A. Sketch the trajectories of the system in the phase plane, indicating the direction of flow, and classify the type and stability of the critical point at the origin. State the equation of any straight line trajectories and find the equation of the line on which the slope of the trajectories is zero. Determine the points in the phase plane for which y/1 is increasing with time. Let y1 = y, and y2 = & then y -y2 and j2 = - 2y1 + 3 y 2 In matrix form (y) = (9 2)/3. -2 3/1y2/ Since det A = 2, trace A = 3 and (trace A) 2- 4det A = 1>0 the critical point is an unstable node. Eigenvalues (-1) (3-2) +2 = 0 => 12 -32+2 =0 (7-2) (2-1) = 0 1=2,1 Eigen vectors 1=2 (A-ZI)(V)=(=2)()=[8) => 2") = (2) 1=1 .. 2(2)=(1) 2€ y =Ge (2) + Ge+(1) Straight line trajectènes C2 == = > 42=24, is a strought line trajectory with exponential growth C =0 yz = y, Page 2 of 9
.. (b) (16 marks) Find all the critical points for the following nonlinear system (*)-( y,y=0 1 -2y13/2 v7 + 31/2 + 41/3- 4 ). Then use linearisation to find the type and stability of the critical points which lie on the y, axis. Critical pts AND y:2+y, y2+4y2-4 = 0 2 1=> y=0 or y2 = 0 If y1=0 in 2 y2 = = 1 ya=0 2 y,