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Unformatted text preview: FINAL EXAMINATION 65.2400 Spring — 1995 NAME: Instructor: Drew Kovacic Ladd Roytburd Good afternoon. Please answer all questions, showing your work clearly and giving reasons for
your answers where appropriate. You may use one 8.5 by 11 sheet of notes while taking the examination. No other reference
material, books, notes, or calculators are permitted. Please be sure to identify your instructor's name in the list above. PROBLEM # POINTS
_ All eight questions have equal weight. {a5 NAME .._____ ______________________ __ 1. (a) Find the solution of the initial value problem d
—y+xy=ac, y(0) =0.
dac (b) Find the general solution of y” + 31/ + 2y = e”. NAME _______ __________________________ _ 2.(a) Find the explicit solution y = (15(23) of the initial value problem dy a: + my2
:v y
(b) Suppose the matrix A has complex eigenvalues n == —3 + 22' and T2 = —3 — 2i and the corresponding eigenvectors e_ 1+3z' 6 _1—32'
1‘ 2—2" 2’ 2+2" Find the real—valued general solution of the system 26’ = Ax. NAME _'__________ ______________ __ 3. (a) A major league baseball team owner has $200 million at the beginning of the 1995
season. The money is invested in the stock market and earns a 10% return. The owner also
pays salaries to the team totalling $3 million per year, where s is a constant. The proﬁt
made by the team is projected to be 5532/ 100 million per year. Let q(t) denote the total wealth (in millions) of the owner at time t.
(1) State (but do NOT solve) an initial value problem for q(t).
(ii) Determine the salary level, or levels, at which the owner’s total wealth remains constant. (iii) For what range, or ranges, of the salary level 3 does the owner’s wealth increase? (b) The solution of a certain initial value problem is
y = 2e‘t — 3te“. State a problem that has this solution. NAME __._____.____._______________________
4. A mass weighing 8 pounds stretches a spring 6 inches. Consider the following possibilities for this springmass system. (a) Suppose that there is no damping and the motion is unforced. The mass is set in motion
from its equilibrium position and is given an initial velocity of 2 inches per second. Set up
and solve the initial value problem that describes the motion of the mass. Also, determine the amplitude and period of the motion. (b) Suppose now that the motion is unforced and that a damping forceis present. Find the value(s) of the damping coefﬁcient, 0, that make the motion critically damped. (C) Now suppose there is no damping and that the motion is forced. Give an example of a
forcing function, F(t), that will produce resonance in the spring—mass system. Explain in words how you know that the function you chose will produce resonance. NAME .________ ____________  5 (a) Find the steady state (equilibrium) solution associated with the modiﬁed heat conduc tion problem 1
um+zu=uh 0<:1:<7r, t>0 u(0,t) = 07 u(7r7 t) 2 10, t > 0 u($,0)=7r—1:, 0<x<7r NAME _________________._______________. (b) Consider the modiﬁed heat conduction equation 1
uzx+1u=uh O<:1:<5, t>0 with the boundary conditions
u(0,t) = 0, ux(5,t) = 0, t > O and the initial conditions u(:c,0) =:r(5—:I:), 0<5r<5 If u(x, t) = X($) T(t)7 ﬁnd ordinary differential equations and boundary or initial conditions for X and TH), respectively. Which equation would you solve first? Why? NAME 0, donor/2 6. Let f(:1:)= 1 7r/2<r<37r/2 0 37T/2<I<2’/T (a) Sketch the graph of the even periodic extension of f for 37r < a: < 37r.
‘1 (b) Find the Fourier series of the extended function. (e) To what value does the Fourier series converge at z = —37r/2? NAME 7. Consider the system due dy
d—t —x<y—2), arm—1) (a) Find all equilibrium solutions (critical points) of this system. (b) Find the linear system that is valid near each critical point. (c) Classify each critical point as to type (node, saddle, spiral, center) and state Whether it is asymptotically stable, stable, or unstable. NAME ___...— _________________________ __ 8. For each of the following equations or systems identify the corresponding phase portrait and explain brieﬂy why you made the choice you did. Note that y = :v’ in the phase portraits When necessary. (a) as" + x = 0 Phase portrait ______________ __
Explanation:
(b) 1‘” — :1: = 0 Phase portrait __________ ___
Explanation:
(c) 33’ = —:c + 3/ Phase portrait ________________
y’ = I * y
Explanation: (d) 55’ = 25 Phase portrait ___________  y’ = 3y
Explanation:
(e) as” —+— 320’ + x — x3 = 0 Phase portrait __._.______
Explanation: 10 PHASE EoRTRMTS Fok PROBLEMS " a
(‘3 / x (2.3 x
\\
. H a .
‘1 ' a ...
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 Spring '04
 Yoon
 Differential Equations, Equations, Boundary value problem

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