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Rutgers | MATH 527
Professors
- Speer
100 sample documents related to MATH 527
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Rutgers MATH 527
The Van der Pol oscillator The Van der Pol oscillator is governed by the second order equation x (1 x2)x + x = 0. To convert this to a system of rst order equations in two unknowns we let y = x and nd x = y, y = x + (1 = x2)y. What follows are phase plane -
Rutgers MATH 527
642:527 ASSIGNMENT 1 FALL 2009 Turn in starred problems Thursday 09/10/2009. Multiple-page homework must be STAPLED when handed in. Section 4.2: 1 (a), (d), *(e), (g), *(k) 2 (b), (d), *(e) *6 3 (a), (e), (i), *(m) 7 (a), *(f), (i), (j) Hints and remarks: -
Rutgers MATH 527
642:527 ASSIGNMENT 2 FALL 2009 Turn in starred problems Tuesday 09/15/2009. Multiple-page homework must be STAPLED when handed in. Section 4.3: 1 (a), (b), (c), *(g), *(n) 2 6 (a), *(j) -
Rutgers MATH 527
642:527 ASSIGNMENT 3 FALL 2009 Turn in starred problems (including problem 3.A) on Tuesday 09/22/2009. DO NOT TURN IN SOLUTIONS FOR ANY UNSTARRED PROBLEMS. Multiple-page homework must be STAPLED when handed in. Section 4.3: 6. (e), (f), (u), *(v) Section -
Rutgers MATH 527
642:527 ASSIGNMENT 4 FALL 2009 Turn in starred problems on Tuesday 09/29/2009. DO NOT TURN IN SOLUTIONS FOR ANY UNSTARRED PROBLEMS. Multiple-page homework must be STAPLED when handed in. Section 5.2: 1(a), (c), (g), (j); 6; 8; 11 Section 5.3: 1 (a), (b), -
Rutgers MATH 527
642:527 ASSIGNMENT 5 FALL 2009 None of these problems will be collected. Section 5.5: 1 (a)(d), 5 (d), 7 (d) Section 5.6: 1 (a), (c), (d), (e), (i), 2 (d), 3 -
Rutgers MATH 527
642:527 ASSIGNMENT 6 FALL 2009 CHANGE IN DUE DATE OF THIS ASSIGNMENT: Turn in starred problems Thursday 10/15/2009. Be sure to read the instructions below. Section 7.2: 1; 4 (a), (b)*; 5 (a) (c) (f); 10 Section 7.3: 1 (a), (c)*, (h); 9 (a), (g), (h)*, (j) -
Rutgers MATH 527
642:527 ASSIGNMENT 7 FALL 2009 Turn in starred problems Tuesday 10/27/2009. Section 7.4: 2 (a), (b)*, (f)* Section 7.4: 7 Section 7.5: 4* Problem 7.A* Two interacting populations x(t), y (t) are described by the equations x = (1 x)x , y = (3 y x)y . See i -
Rutgers MATH 527
642:527 ASSIGNMENT 8 FALL 2009 Turn in starred problems Tuesday 11/03/2009. Section 9.9: 4 (a), (e)* Section 9.10: 2 (a), (c)*; 3 Section 17.2: 5 (a), (e), (g); 12 (a), (e), (j)*, (s) Section 17.3: 1, 4 (a), (c)*, (l)* 8.A* Exercise 1 from the notes on Ex -
Rutgers MATH 527
642:527 ASSIGNMENT 9 FALL 2009 Turn in starred problems Thuesday 11/10/2009. Section 17.3: 4 (g), 16 (b)* Section 17.4: 1 (b), 2 (c)* (d)* (see comment 1 below!) Section 18.3: 6 (c), (h), (n) (See comments 2 and 3 below). 9.A* Do problem 18.3.6(e) but cha -
Rutgers MATH 527
642:527 ASSIGNMENT 10 FALL 2009 No problems from this assignment will be collected. These exercises are just for you to look at simple Sturm-Liouville problems. Some solutions will be posted. Section 17.7: 1 (a), (b), (c), (d), (e), (f); 2 Comments, hints -
Rutgers MATH 527
642:527 ASSIGNMENT 11 FALL 2009 Turn in starred problems Tuesday 12/01/2008. Section 17.7: 7 Section 17.8: 2 (a), (b)*, (d)*, 5 Section 18.3: 6 (i), (l), (m)*, 10 (a), (c)*, (f)*, (i) Comments, hints, instructions: 1. 17.7:7 shows that innocent looking bu -
Rutgers MATH 527
642:527 ASSIGNMENT 12 FALL 2009 This is a short assignment, with only three problems to be turned in. But, as usual, you should work all the listed problems. There will be one more assignment, which will not be collected. Turn in starred problems Tuesday -
Rutgers MATH 527
642:527 ASSIGNMENT 13 FALL 2009 No problems from this assignment will be collected. Section 19.2: 5, 6, 8 Section 19.4: 4, 6 (a), (c) 13.A Consider the following problem for the function u(x, t): 9 uxx = ut , 0 < x < 1, t > 0; u(0, t) = 0, u(1, t) + ux (1 -
Rutgers MATH 527
642:527 ORBITS OF CENTERS AND FOCI FALL 2009 A. Centers Consider a linear system z = A z, with z= x , y A= a c b , d (4.1) for which the origin of the phase plane is a center, that is, for which the eigenvalues of A are pure imaginary. Recall that if p = -
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Rutgers MATH 527
642:527 EXAM 1 SOLUTIONS 3n (x 2)2n converge? n2 n=0 n n+1 2 FALL 2009 1. For what values of x does the power series Solution: We use the ratio test: lim n an+1 3n+1 (x + 2)2n+2 /(n + 1)2 = lim = 3|x + 2|2 lim n n an 3n (x + 2)2n /n2 = 3|x + 2|2 . The s -
Rutgers MATH 527
642:527 SOLUTIONS: EXAM 2 FALL 2009 1. (a) Find the general solution of z = Az, where z = x 4 4 and A = . Be sure you y 1 4 actually give this solution (which should involve two free constants). (b) Give a careful drawing of the phase plane (xy -plane) fo -
Rutgers MATH 527
642:527 FORMULA SHEET FOR EXAM 1 FALL 2009 Taylor series (with radii of convergence given): 1 = 1 + x + x2 + = 1x x xn , 0 |x| < 1 xn , n! |x| < |x| < |x| < x2 x3 e =1+x+ + + = 2! 3! cos x = 1 sin x = x x4 x2 + = 2! 4! x3 x5 + = 3! 5! 0 0 (1)n x2n , -
Rutgers MATH 527
642:527 FORMULA SHEET FOR EXAM 2 FALL 2009 Various Fourier-type expansions: We write f (x) series to indicate that the given series is some Fourier-type expansion of f (x). All the formulas for coecients come from the formula f , n cn = n , n with 1 , 2 , -
Rutgers MATH 527
642:527 FORMULA SHEET FOR FINAL EXAM FALL 2009 Taylor series (with radii of convergence given): 1 = 1 + x + x2 + = 1x ex = 1 + x + cos x = 1 sin x = x xn , 0 |x| < 1 xn , n! |x| < |x| < |x| < x2 x3 + + = 2! 3! 0 x2 x4 + = 2! 4! x5 x3 + = 3! 5! 0 (1)n -
Rutgers MATH 527
642:527 THE HEAT EQUATION IN A DISK Periodic and singular Sturm-Liouville problems FALL 2009 In these notes we study the two-dimensional heat equation in a disk of radius a: 2 2 u(x, y, t) = u(x, y, t), t x2 + y 2 a2 . Here 2 , also written as , is the tw -
Rutgers MATH 527
642:527 METHODS OF APPLIED MATHEMATICS FALL 2009 Instructor: Professor Eugene Speer Hill Center 520, 7324452390, Extension 1313 Web: http:/www.math.rutgers.edu/courses/527/527-f08/ Email: speer@math.rutgers.edu Monday 9:0010:00 AM, Hill 520 Tuesday 3:204: -
Rutgers MATH 527
642:527 PREREQUISITE QUIZ: SOLUTIONS 2n x3n . n2 n=0 Fall 2009 1. Find the radius of convergence of the power series Solution: We use the ratio test: if bn = 2n x3n /n2 is a typical term of the series then n lim 2n+1 x3(n+1) /(n + 1)2 2|x|3 n2 bn+1 = lim -
Rutgers MATH 527
642:527 (16) REVIEW EXAM 1 FALL 2009 1. (a) Find the Laplace transform Y (s) of the solution y (t) of the initial value problem y + 2y + 5y = C (t ), where C is a constant. (b) Find y (t) by taking the inverse Laplace transform of Y (s). (c) Find a value -
Rutgers MATH 527
642:527 REVIEW EXAM 2 FALL 2009 1. (a) Find the general solution of z = Az, where z = x 2 3 and A = . Be sure you y 2 5 actually give this solution (which should involve two free constants). (b) On the axes below give a careful drawing of the phase plane -
Rutgers MATH 527
4 cq Sam p4 Sxm p4 p am q qq \"pS4am p vm % \"pS4tam 4 p am % 4 4 p am vkUUj|q|v!kyQj hi e l e r x r h fi 4 p 4 m l ei h f e ri f r 6 p m qqd!3qrVkjkikUkUFvqQ7 Q|qe r x e hi hi hi Uykekks3Q UUxcfw_ h i f l e h r p m p iki~izijzex7farS4 p p 4 m p m m p -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 2 FALL 2009 Some of these solutions were written by Professor Dan Ocone. 4.3.1 (a) The answer in the text seems sucient. (b) Rewrite the equation as y [(cos x)/x]y +[5/x]y = 0. Then p(x) and q (x) are analytic at all points e -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 3 FALL 2007 4.3.6 (e) This is an Euler equation and one can look for its solution in the form xr without need for the full series. Plugging this into the equation x2 y + xy y = 0 leads to xr (r (r 1) + r 1) = 0. Hence xr will -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 4 FALL 2009 Some of these solutions were written by Professor Dan Ocone. 5.2.1 A function f has exponential order as t if there exist constants K 0, T 0, and a constant c, such that |f (t)| Kect for all t T . (4.1) A straight -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 5 FALL 2009 Some of these solutions were written by Professor Dan Ocone. 5.5.1 (a), (d): See solutions in text. (b) f (t) = H (t)et H (t 1)et . The rst H (t) is in a sense not needed, since in discussing the Laplace transform -
Rutgers MATH 527
<?xml version=\"1.0\" encoding=\"UTF-8\"?> <Worksheet> <Version major=\"12\" minor=\"0\"/> <Label-Scheme value=\"2\" prefix=\"/> <View-Properties presentation=\"false\"></View-Properties> <MapleNet-Properties elisiondigitsbefore=\"100\" labelling=\"true\" indentamount=\"4\" -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 6 FALL 2009 NOTE: A. There is a Maple worksheet on the Assignments and solutions web page which shows the phase plane portraits for some of these problems. I do not know how to have Maple put arrows on curves, so for many of -
Rutgers MATH 527
<?xml version=\"1.0\" encoding=\"UTF-8\"?> <Worksheet> <Version major=\"12\" minor=\"0\"/> <Label-Scheme value=\"2\" prefix=\"/> <View-Properties presentation=\"false\"><Hide name=\"Group Range\"/></View-Properties> <MapleNet-Properties elisiondigitsbefore=\"100\" labelli -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 7 FALL 2009 NOTE: A. There is a Maple worksheet on the Assignments and solutions web page which shows the phase plane portraits for problems 2(b) and 2(k). B. Vectors are denoted either by boldface or, for Greek letters, unde -
Rutgers MATH 527
<?xml version=\"1.0\" encoding=\"UTF-8\"?> <Worksheet> <Version major=\"12\" minor=\"0\"/> <Label-Scheme value=\"2\" prefix=\"/> <View-Properties presentation=\"false\"></View-Properties> <MapleNet-Properties elisiondigitsbefore=\"100\" labelling=\"true\" indentamount=\"4\" -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 8 FALL 2009 Section 9.10: 2 (a) See text solution. (c) With u = (3, 0, 1, 4, 1) we have 1 , u = 5/ 5, 2 , u = 6/ 6, 3 , u = 4, so the best e e e approximation is 3 Section 9.9: 4 (a) See text solution. (e) The formula is (23) -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 9 FALL 2009 Section 17.3: 4 (g) f (x) = | sin x| has period (since sin(x + ) = sin x) so its Fourier series will have the form FS f = a0 + n=1 [an cos 2nx + bn sin 2nx]. Since f (x) is even (| sin(x)| = | sin x| = | sin x|) t -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 10 FALL 2009 Section 17.7:1. In all sections of this problem the equation is y + y = 0; comparing with the standard Sturm-Liouville form (1.a) we see that p(x) = 1, q (x) = 0, and w(x) = 1. The solutions of the dierential equ -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 11 FALL 2008 Section 17.7: 7. Let us rst consider > 0 and then write the solution of the ODE as y = A cos x + B sin x. Then y = A sin x + B cos x the boundary conditions y (0) y (1) = 0 and y (0) + y (1) = 0 become A (A cos + -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 12 FALL 2009 Section 17.10: 2. If Re a > 0 then limx e(ai)x = 0 for real, so F cfw_H (x)eax = 0 H (x)eax eix dx = e(ai)x dx = e(ai)x a i 0 = 1 . a i 3. If a > 0 then all the integrals below converge: F cfw_e a|x| = 0 e a -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 13 FALL 2009 Section 19.2: 5. (a) The wire loop is treated as massless, so any force on it would cause innite acceleration: there can be no (net) force on the loop. A nonzero ux (L, t) would lead to a nonzero vertical force, -
Rutgers MATH 527
642:527:01 APPROXIMATE SYLLABUS FALL 2009 This syllabus is tentative; elements such as scheduling of exams and coverage on each exam may change. All changes will be posted on the class web site. Section numbers refer to Advanced Engineering Mathematics (2 -
Rutgers MATH 527
<?xml version=\"1.0\" encoding=\"UTF-8\"?> <Worksheet> <Version major=\"12\" minor=\"0\"/> <Label-Scheme value=\"2\" prefix=\"/> <View-Properties presentation=\"false\"></View-Properties> <MapleNet-Properties elisiondigitsbefore=\"100\" labelling=\"true\" indentamount=\"4\" -
Rutgers MATH 527
<?xml version=\"1.0\" encoding=\"UTF-8\"?> <Worksheet> <Version major=\"11\" minor=\"0\"/> <Label-Scheme value=\"2\" prefix=\"/> <View-Properties presentation=\"false\"></View-Properties> <MapleNet-Properties warnlevel=\"3\" longdelim=\"true\" plotoptions=\" echo=\"1\" error -
Rutgers MATH 527
642:527 FALL 2009 SUMMARY OF THE METHOD OF FROBENIUS Consider the linear, homogeneous, second order equation: y + p(x)y + q (x)y = 0. Suppose that x = 0 a regular singular point: (1) xp(x) = n=0 pn x , n | x| < R 1 , x q ( x) = n=0 2 qn xn , | x| < R 2 -
Rutgers MATH 527
642:527 EXPANSIONS IN ORTHOGONAL BASES Vector spaces FALL 2009 We will use without much further comment the idea of a vector space. Basically, a vector space is a set of vectors (these vectors will in fact often be functions) with the property that a line -
Rutgers MATH 527
The Predator-Prey System (Lotka-Volterra equations) If x is the prey species and y the predator, then the Lotka-Volterra equations are x = x(a y ), y = y (c + x) where a, c, , and are positive constants. For the graphs that follow we take a = 1.4, c = 2.0 -
Rutgers MATH 527
Methods of Applied Mathematics, 642:527 Guide to Prerequisites I. Prerequisites and entrance into Math 527. The mathematical prerequisites for 642:527, Methods of Applied Mathematics, are the calculus of single variable and multivariable functions, e -
Rutgers MATH 527
642:527 METHODS OF APPLIED MATHEMATICS FALL 2007 Instructor: Professor Eugene Speer Hill Center 520, 7324451313 (Rutgers extension 51313) Web: http:/www.math.rutgers.edu/courses/527/527-f07/ Tuesday 9:009:40 AM, Murray 115 Friday F 9:009:40 AM, Murra -
Rutgers MATH 527
642:527:01 APPROXIMATE SYLLABUS FALL 2007 This syllabus is tentative. An on-line version, accessible through the class web page at http:/www.math.rutgers.edu/courses/527/527-f07/ will be updated periodically throughout the semester. Section number -
Rutgers MATH 527
642:527 THE HEAT EQUATION IN A DISK Periodic and singular Sturm-Liouville problems FALL 2007 In these notes we study the two-dimensional heat equation in a disk of radius a: 2 2 u(x, y, t) = u(x, y, t), t x2 + y 2 a 2 . Here 2 , also written as -
Rutgers MATH 527
Review problems: Math 527, Exam 2 Problems 16 are from last year\'s second midterm exam; problem 7 is additional. 1. Consider the system x = 1 - xy, y = x - y 3 . Determine its singular (equilibrium) points and classify each, insofar as possible, usi -
Rutgers MATH 527
Eigenvalues and eigenfunctions of a Sturm-Liouville problem In class on November 13 we discussed the problem y + y = 0 for 0 < x < 1; y(0) = 0, y (1) - y(1) = 0. We found the eigenvalues to be 1 = 0, n = 2 for n 2, where n is the (n - 1)st positiv -
Rutgers MATH 527
s := N -> 1/2+sum(2*sin(2*k+1)*x)/(2*k+1)*Pi),k=0.N-1); s(3); plot([s(0),s(1),s(3),s(20),s(50)],x=0.2*Pi,color=[RED,GREEN, MAGENTA,BROWN,BLUE],thickness=1,numpoints=500); b := n -> (10-n^2)/(n^4-16*n^2+100): a := n -> -2*n/(n^4-16* n^2+100): y := -
Rutgers MATH 527
Competing Species Models The general equations for competing species are of the form x = x(1 1x 1y) y = y(2 2y 2x), where x and y are the populations of two species in some environment, 1 and 2 are positive constants representing the growth rat -
Rutgers MATH 527
The Nonlinear Pendulum The nonlinear pendulum equation is g = - sin - , l where is the angle that the pendulum makes from a downward vertical axis, measured counterclockwise; g is the gravitational constant, l the length of the pendulum, and -
Rutgers MATH 527
Methods of Applied Mathematics Math 527:01 1. Find the power series expansion (Taylor series) centered at x0 = 1 for radius of convergence. 2. (a) Find the Laplace transform of f (t) = (b) Find the inverse transform of 0, if 0 t < 1; e2t , if t 1. -
Rutgers MATH 527
642:527 FALL 2007 SUMMARY OF THE METHOD OF FROBENIUS Consider the linear, homogeneous, second order equation: y + p(x)y + q(x)y = 0. Suppose that x = 0 a regular singular point: (1) xp(x) = n=0 pn x , n |x| < R1 , x q(x) = n=0 2 qn xn , -
Rutgers MATH 527
642:527 METHODS OF APPLIED MATHEMATICS FALL 2008 Instructor: Professor Eugene Speer Hill Center 520, 7324451313 (Rutgers extension 51313) Web: http:/www.math.rutgers.edu/courses/527/527-f08/ Email: speer@math.rutgers.edu Monday 1:403:00 PM, Hill 520 -
Rutgers MATH 527
642:527:01 APPROXIMATE SYLLABUS FALL 2008 This syllabus is tentative; elements such as scheduling of exams and coverage on each exam may change. All changes will be posted on the class web site. Section numbers refer to Advanced Engineering Mathem -
Rutgers MATH 527
Methods of Applied Mathematics: Math 527:01 Final Exam, Fall 2006 Instructions: You must show all work to earn full credit. If you do not have room in the given space to answer a question, use the back of one of the exam pages and indicate clearly -
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Rutgers MATH 527
642:527 THE HEAT EQUATION IN A DISK Periodic and singular Sturm-Liouville problems FALL 2008 In these notes we study the two-dimensional heat equation in a disk of radius a: 2 2 u(x, y, t) = u(x, y, t), t x2 + y 2 a 2 . Here 2 , also written as -
Rutgers MATH 527
642:527 EXPANSIONS IN ORTHOGONAL BASES Vector spaces FALL 2008 We will use without much further comment the idea of a vector space. Basically, a vector space is a set of vectors (these vectors will in fact often be functions) with the property that -
Rutgers MATH 527
Slope field, nullclines, and trajectories: predator-prey model 6 5 4 y 3 2 1 0 1 2 x 3 4 5 Predator-prey populations 5 4 3 x(t) 2 1 0 0.0 2.5 5.0 t 7.5 10.0 Predator-prey populations 6 4 y(t) 2 0 0.0 2.5 5.0 t 7.5 10.0 -
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Rutgers MATH 527
642:527 PREREQUISITE QUIZ: SOLUTIONS x2 e-x dx 0 Fall 2008 1. Does converge or diverge? Justify your answer. Solution: The function is finite for all x so the only question is convergence at x = . Intuitively, we get convergence because, as x -
Rutgers MATH 527
d i U P H X Wdd g r Q W b ` Q g tS Q g ` Q g Wd ` g W w P W t X W P H w Rg 5YVfT6Rc2sVv5FhVaiqRqTYY&H Q uS U g X W g b H ` U W W ` P g X g u t W X RVRc2hv6cqqTliv2cig n VqVcT4I5RTVd zz4zFzs PS Q WS X W Q X W H f -
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Rutgers MATH 527
642:527 SOLUTIONS: EXAM 1 FALL 2008 1 (a) Find the Laplace transform Y (s) of the solution y(t) of the initial value problem y + 2y + 5y = C(t - ), y(0) = 0, y (0) = -3, where C is a constant. Solution: Taking the Laplace transform of the equa -
Rutgers MATH 527
642:527 ASSIGNMENT 10 FALL 2008 No problems from this assignment will be collected. These exercises are just for you to look at simple Sturm-Liouville problems. Some solutuions will be posted. Section 17.7: 1 (a), (b), (c), (d), (e), (f) Comments, -
Rutgers MATH 527
642:527 (20) REVIEW EXAM 2 FALL 2008 x 4 3 and A = . y 1 2 (b) Give a careful drawing of the phase plane (xy-plane) for this system, showing enough trajectories to indicate qualitatively the motion in each region of the plane, as well as any \"spec -
Rutgers MATH 527
642:527 SOLUTIONS: EXAM 2 FALL 2008 x 2 3 and A = . Be sure you y -2 -5 actually give this solution (which should involve two free constants). (b) Give a careful drawing of the phase plane (xy-plane) for this system, showing enough trajectories to -
Rutgers MATH 527
642:527 ASSIGNMENT 13 FALL 2007 This assignment will not be collected. The purpose is to give you some problems on topics not covered in the earlier, collected, assignments. Inhomogeneous boundary value problems: Section 18.3:6 (a), (i), (l), (m) -
Rutgers MATH 527
642:527 ASSIGNMENT 12: REVISED FALL 2007 Multiple-page homework must be STAPLED when handed in. Turn in starred problems Thursday 12/6/2007. Problems marked with two stars will be treated as extra credit.problems Section 17.7: 8 Section 18.3: 10 ( -
Rutgers MATH 527
\' w k x g q k gm j n \" g m h fiifzPSor0edi{ iSfP#iSePeergPePeTif h k g g q g q k l | m x g k h n gm l k j g q gm l k k j g n k h g m g q k g m h g q g m w \' w \' g j k m -
Rutgers MATH 527
642:527 ASSIGNMENT 10 FALL 2007 Multiple-page homework must be STAPLED when handed in. Turn in starred problems Tuesday 11/13/2007. Section 17.3: 4 (g), 16 (b)*, 18 (c)* Section 17.4: 1 (b), 2 (c)* (d)* (see Comment 2 below!) Section 18.3: 6 (c), -
Rutgers MATH 527
642:527 ASSIGNMENT 9 FALL 2007 Multple-page homework must be STAPLED when handed in. Turn in starred problems Tuesday 11/6/2007. Section 9.9: 2 (a), (d)*; 12 (d), (f)* Section 9.10: 2 (a), (c)*; 3 Section 17.2: 5 (a), (g), (f); 12 (a), (e)*, (j)*, -
Rutgers MATH 527
642:527 ASSIGNMENT 8 FALL 2007 Multple-page homework must be STAPLED when handed in. Turn in starred problems Tuesday 10/30/2007. Section 7.4: 7 (a)*, (b), (c) Section 7.5: 4*; 6 Section 9.6: 11; 12 (b), (c), (d)*, 13* Problem 8.A* Two interacting -
Rutgers MATH 527
642:527 ASSIGNMENT 7 FALL 2007 Multple-page homework must be STAPLED when handed in. Turn in starred problems Tuesday 10/23/2007. Section 7.4: 2 (a)*, (b)*, (k)* In each of these problems, do the following: Find all singular points; Obtain the l -
Rutgers MATH 527
642:527 ASSIGNMENT 6 FALL 2007 This homework will not be collected. Turn in starred problems Tuesday 10/16/2007. Be sure to read the instructions below. Section 7.2: 1; 4 (a), (b)*; 5 (a) (c) (f); 10 Section 7.3: 1 (a), (c)*, (h); 9 (a), (g), (h)* -
Rutgers MATH 527
642:527 ASSIGNMENT 5 FALL 2007 This homework will not be collected. Section 5.5: 1 (a)(d), 7 (d), (f) Section 5.6: 1 (a), (c), (d), (e), (i) 1 -
Rutgers MATH 527
642:527 ASSIGNMENT 4 FALL 2007 Multple-page homework must be STAPLED when handed in. Turn in starred problems Tuesday 010/02/2007. Section 5.2: 1(a), (c), (g), (j); 6*; 8*; 11 Section 5.3: 1 (a), (b), (c)*; 3 (a), (c); 5 (a); 10 (a), (c)*, (e) Sec -
Rutgers MATH 527
642:527 ASSIGNMENT 3 FALL 2007 Multple-page homework must be STAPLED when handed in. Turn in starred problems Tuesday 09/25/2007. Section 4.5 8 9* Section 4.6 1 2* (see Remark 2 below) 6* 12 (a), (c)*, (d), (e)* 15 Remarks: 1. Problem 4.5.9 -
Rutgers MATH 527
642:528 ASSIGNMENT 2 FALL 2007 Turn in starred problems Tuesday 09/18/2007. The problems from 4.2.6 below illustrate the various possibilities for the Frobenius method: all three cases of Theorem 4.3.1 occur, and for case (iii) there is an example -
Rutgers MATH 527
642:528 ASSIGNMENT 1 FALL 2007 Turn in starred problems Tuesday 09/11/2007. I have listed 4.2.6 before 4.2.3 because 4.2.6 teaches us an important lesson about finding Taylor series: it is frequently easiest to get a Taylor series starting from so -
Rutgers MATH 527
642:527 ASSIGNMENT 1 FALL 2008 Turn in starred problems Tuesday 09/09/2008. Multiple-page homework must be STAPLED when handed in. I have listed problem 4.2.6 before problem 4.2.3 because 4.2.6 teaches us an important lesson about nding Taylor ser -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 1 FALL 2008 Some of these solutions were written by Prof. Dan Ocone. 4.2 Rather than using (7a) or (7b) it is usually better use the ratio test directly. (n + 1)xn+1 n+1 1. (a) Since lim = |x|, the series 0 nxn conve -
Rutgers MATH 527
642:527 ASSIGNMENT 2 FALL 2008 Turn in starred problems Tuesday 09/16/2007. Multiple-page homework must be STAPLED when handed in. The problems from 4.2.6 below illustrate the various possibilities for the Frobenius method: all three cases of Theo -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 2 FALL 2008 Some of these solutions were written by Professor Dan Ocone. 4.3.6 (a) The equation is 2x2 y + xy + x4 y = 0. (We have multiplied through by an additional factor of x to simplify bookkeeping.) Substitut -
Rutgers MATH 527
642:527 ASSIGNMENT 3 FALL 2008 Turn in starred problems (and only those) on Tuesday 09/23/2007. Multiple-page homework must be STAPLED when handed in. Section 4.5 8 9* Section 4.6 1 2* (see instructions in Remark 2 below) 5 (a)* 6 12 (a), ( -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 3 FALL 2007 Some of these solutions were written by Professor Dan Ocone. 4.5.8 The solutions in the text show how to do these. 9. In the denition (14) of the Gamma function we make the change of variable t = 2u2 , dt -
Rutgers MATH 527
642:527 ASSIGNMENT 4 FALL 2008 Turn in starred problems Tuesday 09/30/2008. Section 5.2: 1(a), (c), (g), (j); 6; 8; 11 Section 5.3: 1 (a), (b), (c); 3 (a), (c); 5 (a); 10 (a), (b)*, (e) Section 5.4: 1 (a), (b), (d), (g), (i)*, (o)*, (t) Section 5. -
Rutgers MATH 527
642:527 ASSIGNMENT 5 FALL 2008 None of these problems will be collected. Section 5.5: 1 (a)(d), 5 (d), 7 (d) Section 5.6: 1 (a), (c), (d), (e), (i) 1 -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 5 FALL 2008 Some of these solutions were written by Professor Dan Ocone. 5.5.1 (a), (d): See solutions in text. (b) f (t) = H(t)et H(t1)et . The rst H(t) is in a sense not needed, since in discussing the Laplace trans -
Rutgers MATH 527
642:527 ASSIGNMENT 6 FALL 2008 Turn in starred problems Tuesday 10/14/2007. Be sure to read the instructions below. Section 7.2: 1; 4 (a), (c)*; 5 (a) (c) (f); 10 Section 7.3: 1 (a), (e)*, (h); 9 (a), (b)*, (d)*, (g); 11 (d), (g)* Instructions: In -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 6 FALL 2008 NOTE: A. There is a Maple worksheet on the Assignments and solutions web page which shows the phase plane portraits for some of these problems. I do not know how to have Maple put arrows on curves, so for -
Rutgers MATH 527
642:527 ASSIGNMENT 7 FALL 2008 Since I was late posting this assignment, I amc changing the due date to Thursday. Turn in starred problems Thursday 10/23/2007. Section 7.4: 2 (a), (b)*, (k)* Section 7.4: 7 Section 7.5: 4* Problem 7.A* Two interact -
Rutgers MATH 527
642:527 SOLUTIONS: ASSIGNMENT 7 FALL 2008 NOTE: A. There is a Maple worksheet on the \"Assignments and solutions\" web page which shows the phase plane portraits for problems 2(b) and 2(k). B. Vectors are denoted either by boldface or, for Greek let
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