~UTS. University of Technology, Sydney TO BE RETURNED AT THE END OF THE EXAMINATION. THIS PAPER MUST NOT BE REMOVED FROM THE EXAM CENTRE.

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1 Cover Type B ~UTS University of Technology, Sydney TO BE RETURNED AT THE END OF THE EXAMINATION. THIS PAPER MUST NOT BE REMOVED FROM THE EXAM CENTRE. SURNAME: FIRST NAME: STUDENT NUMBER: COURSE: AUTUMN SEMESTER 2012 SUBJECT NAME INTRODUCTION TO LINEAR DYNAMICAL SYSTEMS SUBJECT NO DAY/DATE Wednesday 27 June 2012 TIME ALLOWED START/END TIME : Three hours plus ten minutes reading time 9:30am I 12:40pm NOTES/INSTRUCTIONS TO CANDIDATES: Attempt all questions All questions are of equal value Non Programmable Calculators MAY be used Answer each question in a separate booklet CLEARLY indicate the question number on the front of the booklet Examiner: Dr S M Woodcock Assessor: Dr B J Moore

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3 Page 1 Question 1 (( ) + (2+2+2) + ( ) = 25 Marks) (a) LetA=( 2-4 I,B=( 3 0 \c=( JandD=(-5 4). -1-1) 0 2) Calculate each of each of the following: When no answer is possible, briefly explain why. A-B; DtD; WB; (iv) DA; (v) BD; (vi) CDt (vii) det( A )+det( B); (viii) (A+ Br 1 ; (ix) A 1, A 2, the eigenvalues of A. ~) Consider solvillg fue mauix equation l ~ ~ j(; l J ~ i J to find the values of x and y. For which values of p and q does the equation yield: unique solutions; no solutions; non-unique solutions? (c) Let z = re;e = r( cos e + i sin B). By considering both z and its conjugate z = re-;e, show that cos sin 2 8 = 1. Let x =sin( B). Applying de Moivre's Theorem, (COS 8 + i sin 8)" = COS n8 + i sin n8 for n E 2:/ show that sin( 58) = 5x- 20x x 5 Hint: Hence or otherwise find all five values of e E [0, 2n] such that sin( 58)= 18 sin 5 (B) -36sin 3 (8) + 19sin(B). You may find the following Mathematica output helpful. In[1] := Factor[x"5-8x"3 + 7x] Out[1]= (-1 + x) x (1 + x) (-7 + x"2) Over...

4 Page2 Question 2 ((2+3+3) + (2+2+4) + (2+2+5) = 25 marks) (a) Calculate both first partial derivatives (i.e. with respect to x and y) of the following functions: (1.) J( ) ~Y + 12e2x x,y = ' y g(x,y) =sin(.,jx; + ylog(x)) h(x,y) = /e Y (b) A particle's movement in the x-y plane is described by the differential equation Let the eigenvalues of A bea 1, A 7 Assume (x(o)j =1=- (OJ. - y(o) 0 For each of the following cases, characterise the particle's trajectory as stationary, periodic, convergent or divergent. Where the orbit is periodic, state its period; where it is convergent, state to which point it converges: ( 1 13j A 1 = i, A 2 = -1-2i ; A 1 = 3 + 7i, A 2 = 3-7i; A = _ 2 _ 1 ) (c) LclB ~ [ ~6 i2 7] Find a matrix M which diagonalises B. That is, find M such that J = M- 1 BM is diagonal. (Note: You do NOT need to perfonn any calculation to verify that J is diagonal). Write down the matrix exponential of J, exp( J ). Solve the system of equations [x(o)j [OJ x'(t)=-6x+2y-5z y'(t) = x + y + z with initial conditions y(o) = 0 z'(t)=8x-2y+7z z(o) 1 to find the values of x,y and z. Hint: You may find the following Mathematica output helpful. In[1] := b={ {-6,2,-5},{1,1,1 },{8,-2, 7}}; ln[2] := b//eigensystem Out[2]= {{2, -1, 1}, {{-1, 1, 2}, {-1, 0, 1}, {-1, -1, 1}} Over...

5 Page 3 Question 3 ((2+3+2) + (1+2+2) ( ) = 25 Marks) (a) Solve the following differential equations to find y(x) in its most general fonn: y"-y'-42y=2l; y"+9y=16sin(x); y"+4y'+4y=o (b) L x J is the floor function, which for any x E lr returns the value of the largest integer :::;x. Sketch the function f(x)=x+lsin(m:)j on the domain [0,6]. Hence find the first derivativej'(x). Clearly state any points where the function f(x) is non-differentiable. Is f(x) invertible? Justify your answer. (c) The exponential function is defined by the power series zz z z z z exp(z) =e" = ! 2! 3! 4! 5! 6! To which function does the series 2 2x 2x 2 e2xx3 2x 4 e2xx5 e2xx6 4 e x e x e x 7x e- --1-!-+ll- 2x3! + 22 x4! 23 x5! + 24 x6! -... converge? (d) Let g( x) = x3-2x + 2 with domain lr. Find g' ( x) and show that the function has stationary points at x = J2i3 and x = -J2!3. Sketchthegraphof g(x) ontheinterval [-3,3]. Estimate the value of the real root of g(x) = 0 via the Newton Raphson method: g(x) 11 X =Xn+I n '( ) g XII (iv) with starting point x 0 = Give your answer correct to three decimal places. Show that if the starting point x 0 = 1.5 is used, the method fails to converge to the root and instead returns values which are cyclic. State the period of this cycle. (v) Sketch on your graph of g(x) the behaviour of the Newton-Raphson method for starting point x 0 = Over...

6 Page4 Question 4 ((4+3) + ( ) + (3+2+4) = 25 Marks) (a) 1! Consider the problem of evaluating fo2 sin" (x)dx for n E z+. (b) (iv) (v) By writing sin" (x) = sin(x) x sin"- 1 (x) and integrating by parts, show!:. " n-1!:. " thatf2 sm (x)dx = --f1 sm (x)dx. o 11 o!:_!:_ 2048 Evaluate i 2 sin(x)dx and hence show thatf2 sin 15 (x)dx = --. Jo Solve the first order difference equation Un+l = -3U" with initial condition U 0 = 2 to find the value ofuk for k :2::0. Hence or otherwise show thatu 9 = cc d <Xl Let y(x) = L, a 11 x" and hence 2 = L, na 11 x"- 1 Suppose y(x) is a 11=0 dx 11=0 solution of (1 + 3x) dy + 3 y = 0. Show that the coefficients in the series dx expansion of y(x) satisfy a = -3a". Hence find a series solution to the first order differential equation dy -3y - = --withy(o) = 2. dx 1+3x Considering the properties of geometric series, find for which values of x the solution converges to a finite value. (c) Evaluate each of the following definite integrals: f./3 x(x 2-2) 17 dx;.fi SOME STANDARD RESULTS f d x = ~1-x arcsin x + 2 C f d x = arctan x + C 1 + x 2 End ofpaper