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# FinalFall95 - FINAL EXAMINATION 65.2400 NAME Instructor...

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Unformatted text preview: FINAL EXAMINATION 65.2400 December 14, 1995 NAME: Instructor: Boyce Herron Kapila Kovacic Please circle your instructor's name in the list above. Please answer all questions, showing your work in detail. One two-sided sheet of notes is permitted, but no other notes, books, references, calculators, etc. are allowed. PROBLEM # POINTS - _ NAME _______________________________ _- SECTION ___________________ __ 1. (a) Find the solution of the initial value problem d x—y—2y=:r2+1, y(l)=1. dl‘ (b) Consider the initial value problem dr 1 2 —- = -— ' = 1 d6 2r sm 0, r(0) Find the solution 7" = f ((9) and show that it becomes unbounded as 0 approaches a certain ﬁnite value. NAME _______________________________ __ SECTION _____________ __ 2. (a) Consider the differential equation dN — = —N. dt N(2 ) Find the equilibrium solutions. On the axes below sketch the equilibrium solutions and also the solutions passing through the points (0,3) and (O, a respectively. N 3 2. (b) Find the general solution of y” + 4y’ -+- 43/ = 36’2‘. NAME _______________________________ _- SECTION ____.____.____-- 3. Suppose that y = y(t) satisﬁes y” + 2y’ — 3y = 0, y(0) = A, y’(0) 2 ~41, where A is a. constant. (8.) Find y(t). (b) Find A so that y(t) remains bounded as t —> oo. NAME _______________________________ __ SECTION ___________________ __ 4. (a) For large t the solution u(t) of the equation I! 1/ l u +511 +Zu— Sin approaches a steady oscillation, Find the period and amplitude of this eventual steady oscillation. (b) An oscillating system has the phase plane trajectory shown below for 0 S t g 20. Sketch the corresponding graph of 1: versus t. NAME _______________________________ __ SECTION ____________________ __ . 0, O<x<10 O‘Letf(x)‘{25, 10<x<20 (a) Draw the graph of the even extension of f(;z:) of period 40 for ~60 < :r < 60. (b) Find the Fourier series for this extended function. NAME _______________________________ __ SECTION ___________________ __ (0) Write out explicitly the ﬁrst four nonzero terms of the series in part (b). (d) To What value does the series converge when x = 10? NAME _______________________________ __ SECTION _______________ __ 6. Consider the heat conduction problem an = U: 0<I<20, t>0 u(0,t) = 50 t > 0 u(20,t) = 10 t> 0 u(x0)_ 0, 0<\$<10 ’ ‘ 25,10<:z<20 (a) Find the steady state solution 11(27) associated with this problem. (b) Assuming that u(:1c, t) = w(:r, t) +v(\$), write down the partial differential equation7 boundary conditions, and initial conditions that the transient w(:v, t) must satisfy. (C) Assuming that w(z, t) = X T(t), determine the ordinary differential equation and boundary conditions that X must satisfy. NAME _______________________________ __ SECTION _______ __ (d) For the problem originally stated, sketch on the axes below the initial temper- ature distribution, the ﬁnal (as t —> '30) temperature distribution. and the temperature distribution at two intermediate times. (e) Consider the problem X” + 0X = 0, X(0) = O, X’(20) = 0. Assuming that a > 0, ﬁnd the eigenvalues an and eigenfunctions Xn(2:). NAME _______________________________ __ SECTION _________-_____-___ 7. (a) Find the general solution of the linear system dx dr CT; = —5\$1 + 412., Tit—2 = 8551— 1E2- (b) A certain matrix A has eigenvalues r1 = 3 and r2 = —9, with the corresponding 1 eigenvectors {(1) = and £9) = < 1). On the axes below sketch several typical trajectories of z’ = Am in the x1 :rg-plane. (c) On the axes below sketch the graphs of \$1 versus t and 362 versus t for the solution of part (b) that passes through the point (1,0) when t = 0. NOTE. Do not compute anything in part Just draw graphs that are consistent with those in part Be sure to label which graph is which. X7. )‘IJXL Part (L) {Md ((3 NAME _______________________________ _- SECTION ______________ ___ 8.(a) Suppose that the following system is satisﬁed by the populations of two species 13(t) and y(t): dI dy __ = -2 _ = — . 1 dt 2:, dt 9(3 y) ( ) One of the following statements is true; circle that statement. (i) Both species will surely die. (ii) One species may survive but the other will surely die. (iii) Both species may survive. (b) Now consider the modiﬁed system: dx dy 2 Eg——2z+ry, 5-1/(3 y)—ry (2) Find all of the equilibrium points. (C) For one of the equilibrium points both x and y are positive. Find the linear system that approximates Eqs. (2) near this equilibrium point. NAME _______________________________ __ SECTION ____.______--_______- (d) For the linear system in part (c) determine whether the critical point is asymp- totically stable, stable, or unstable, and classify it as to type (saddle, node, spiral, or center). (e) What do the results of parts (0) and (d) imply about solutions of the nonlinear system (2)? ...
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