# FinalFall94 - FINAL EXAMINATION 65.2400 NAME Instructor...

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Unformatted text preview: FINAL EXAMINATION 65.2400 December 20, 1994 NAME: Instructor: Boyce Kovacic Ladd Siegmann Caprioli Roytburd Herron Good morning! Please circle your instructor's name in the list above. Please answer all questions, showing your work in sufficient detail, and giving reasons where they are requested. You may use one (two-sided) 8.5 by 11 sheet of notes in your own handwriting, but no other aids (books, notes, calculators, ...) are allowed. PROBLEM # POINTS NAME _________._________-______-.________ 1. (a) Find the general solution of y’ — 2y = —1+e_x. Also ﬁnd the solution that satisﬁes the initial condition y(0) = yo. Describe how solutions behave as a: ——+ 00 for different values of go. (b) Find the explicit solution y = (MI) of the initial value problem dy ~ 1 a = -(2x -1)y27 9(0) = -- and state the interval in Which the solution exists. NAME .._______..__________-________ 2. (a) For each of the plots shown below ﬁnd an equation of the form dy/dt = f (y) whose solutions have graphs such as those shown. The equations may be either linear or nonlinear. (b) The position u(t) of a certain spring—mass system satisﬁes the equation mu" + 3U = 0. Find the value of the mass m for which the motion of the mass has period four. NAME __..___________--_________..--_____ (c) A certain spring-mass system satisﬁes the equation (1/2)u” + (2/3)”!1’ + ku = 0. Find the value of the spring constant k for which the system is critically damped. NAME _.____.__ ____________________ _-._ 3. (3) Find the solution of the initial value problem 4y”+4y'+y = 0. (b) Find the general solution of y” -+- 4y = 3 cos 2t. NAME _.__.____________________- 4. For each of the equations in the ﬁrst column identify the corresponding graph or phase portrait from the second column. In each case also give a brief statement of your reason. Recall that a single equation can be changed to a system by letting x1 = y, :52 = y’. X1 (a) ”— §y’+4y=0 Graph___.____ Reason (b) y”—y’-2y=0 Graph______ Reason NAME ______._.._-._______________-_______ (Le x mm: 5. Consider the Fourier series f (1:) = 7)— + 2 an cos T. 71:1 1.4 1 2 where an = i /0 :rcos n—ngx (a) Sketch the graph of y = f(r) for —8 < :r < 8. (b) Evaluate a0 and an for n 2 1. NAME _..._.___________________________ (c) Write down a heat conduction problem (differential equation, boundary conditions, and initial conditions) for which this Fourier series is required as part of the solution. Do g); solve the heat conduction problem. NAME __.___.._____________,_,___________--- 6. (a) Consider the damped wave equation 1122 = at: + kut: where k is a positive constant, with the boundary conditions u(0. t) = 0. u(L., t) = 0 and the initial conditions u(J:. 0) = J:(L — I), ut(\$, 0) = 0. Assuming that u(:c, t) = X (x)T(t), ﬁnd ordinary differential equations satisﬁed by X and T(t). Which of the two ordinary differential equations should you solve first? What boundary or initial conditions are associated with this equation? Do M solve the equations that you obtain. NAME ..___..___-____________-‘_-________ (b) Find the eigenvalues and eigenfunctions of the problem X” + 0X = 0, X(0) = 0, X’(10)= 0. You may assume that a > O. NAME ______._____-.____-___,____-_____--__ 7. Consider the system :r’=;r(4—2I+y) y'=y(3—3y+r) (a) Find all of the equilibrium solutions (critical points). (b) Let P be the critical point in the ﬁrst quadrant (both coordinates positive). Find the linear system that approximates the given nonlinear system near P. 10 NAME _______._______________________--- (0) Determine the eigenvalues of the linear system in part (b). (d) Sketch the trajectories of the original nonlinear system near P. 11 NAME ..__.._~___________-______________ 8. A direction ﬁeld for a certain system of the form 23'=F(I~y)~ y’=G(I7y) is shown on the next page. (a) Write down the (approximate) coordinates of each equilibrium solution (critical point). (b) For each critical point determine from the plot whether it is asymptotically stable, stable, or unstable, and classify it as to type (saddle point, spiral point, node, or center). (c) Sketch the approximate extent of the basin of attraction for each asymptotically stable critical point. 12 +-+-n~&‘\'\‘\'\'\'\\'\'\\\'\ 7//_.\‘\‘\‘\ ‘ / (/KW1\'\'\'\'\'\'\'\‘\'\\\\‘—/ / /////KK¢—+—4~n\v\\\\v\v\qu_// J ...
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