18.03 Final Examination 9:00–12:00, May 18, 2010 Your Name Recitation Leader Recitation Time Do not turn the page until you are instructed to do so. Write your name, your recitation leader’s name, and the time of your recitation. Show all your work on this exam booklet. When a particular method is requested you must use it. No calculators or notes may be used, but there is a table of Laplace transforms and other information at the end of this exam booklet. Point values (out of a total of 360) are marked on the left margin. The problems are numbered 1 through 10. 1 2 3 4 5 6 7 8 9 101. (a) and (b) concern the tritium that is leaking from the Oyster Creek Nuclear Generating Station into the aquifer in New Jersey at a certain rate, which we will assume is one kilogram per year. The half life of tritium is 12 years. [6] (a) Ignoring other effects (other sources or sinks of the tritium), set up a differential equation for the amount of tritium in the aquifer as a function of time. (For full credit, determine any constants in the equation.) [6] (b) If this leak goes on for a long time, what will the tritium load in t he aquifer be? (How many kilograms?) dy y2 Pa rt s (c)–(g) of this problem will concern the differential equation = x − . dx 4 [6] (c) Let y(x) denote the solution to this equation such that y(1) = 0. Use Euler’s method with step size 12 to estimate y(2).dy y2 1. Continuing with = x − dx 4 [3] (d) Sketch the isoclines for slopes m = −1, m = 0 , and m = 1, on the plane below. [3] (e) On the same plane, sketch the graph of the solution y(x) with y(1) = 0. [6] (f) For this same solution, suppose that y(x) achieves a minimum at x = a. What is y(a) (in terms of a)? [6] (g) Estimate the value of y(100).2. In (a)–(c) we consider the autonomous equation ˙x = x3 − x2 − 2x. [10] (a) On the vertical line below, sketch the phase line of this equation. [10] (b) Sketch the graphs of some solutions. Be sure to include at least one solution with values in each interval above, below, and between the critical points. [6] (c) Suppose x(0) is quite small, say 0.1. For t > 0 , x(t) is best approximated by (0.1)eat for what value of a? dy sin(x)[10] (d) Solve the initial value problem x + 2y = − , y(π) = 0. dx x� � [6] 3. (a) Find a complex number r (expressed as r = a + bi with a, b real) and a positive real eiω t � π� number ω such that Re = 2 cos 2t − . r 4 [6] (b) Express the cube roots of −8i in the form a + bi (with a and b real). [8] (c) What is the pseudoperiod of a nonzero solution to ¨x + 2 ˙x + 10x = 0?[8] 3. (continued) (d) At what circular frequency ω = ωr does the sinusoidal solution to x¨ + 2 ˙x + 10x = cos(ωt) have the largest amplitude? [8] (e) At what circular frequency ω = ωp is the phase lag of the solution to x¨+2 ˙x+10x = cos(ωt) equal to π 2 ?4. Let p(D) = D2 + 4D + 8I. [10] (a) Find one solution to p(D)x = 8t2 . [10] (b) Find one solution t o p( D)x = e−t . In (c) and (d), suppose that x = t3 is a solution to p(D)x = q(t) (where still p(D) = D2 + 4D + 8I). [8] (c) What is q(t)? [8] (d) What is the solution to p(D)x = q(t) such that x(0) = 0 and ˙x(0) = 2?[15] 5. (a) Find a periodic solution to ¨x + ωn 2 x = sq( 2 t) if one exists. [6] (b)–(c) concern the function f (t), periodic of period 2π, with f(t) = |t| −π 2 (b) Is f(t) even, odd, or neither? What is its average value? Graph f(t). for |t| ≤ π. [15] (c) What is the Fourier series of f(t)?� �� � ��� �� � ��� �� � ��� �� � ��� �� � ������ ��6. (a)–(b) The function f(t) is defined by the following graph, in which the hashmarks are at unit spacing. [6] (a) Graph the generalized derivative f′(t). [8] (b) Write a formula for f′(t), using step and delta functions as necessary. Identify in your formula the regular part fr′ (t) and the singular part fs′ (t).6. (continued) Suppose that the unit impulse respo nse of a certain operator p(D) is w(t) = u(t)e−t sin(2t). [8] (c) Please calculate the solution to p(D)x = u(t)e−t, with rest initial conditions. [6] (d) What is the solution to p(D)x = δ(t − 1) with rest initial conditions? [8] (e) What is the characteristic polynomial p(s)?7. (a)–(c) concern the operator p(D) = D3 + D. [8] (a) What is the transfer function of the operator p(D)? [10] (b) What is the unit impulse respo nse of this operator? [10] (c) What is the Laplace transform X(s ) of the solution to p(D)x = cos(2t) with rest initial conditions? (s + 1)e−s [8] (d) Sketch the pole diagram of s3 + 2s2 + 2s� � � � � � 5 −1 8. In (a) a nd (b), A = . −1 5 [8] (a) What ar e the eigenvalues of A? [8] (b) For each eigenvalue, find a nonzero eigenvector. (c)–(e) concern a certain 2 × 2 matrix B, whose eigenvalues are known to be 2 and −2. [6] (c) What is the trace of B? What is the determinant of B? 1 3 [8] (d) Suppose also that is an eigenvector for value 2 and is an eigenvector for 2 4 value −2. Calculate eBt . � � 1 [6] (e) What is the solution to ˙u = Bu with u(0) = ? 1� � 1 2 9. Let A = , and consider the homogeneous linear system ˙u = Au. −2 a [6] (a) On the (Tr,Det) plane below, sketch the critical parabola (where the matrix has repeated eigenvalues) and sketch line corresponding to the matrices A a s a varies. [30] (b) For each of the following conditions, indicate the values of a (if any) f or which the phase portrait satisfies the condition. (i) Saddle (ii) Stable node = nodal sink (iii) Unstable node = nodal source (iv) Stable spiral = spiral sink (v) Unstable spiral = spiral source (vi) Stable defective node = defective nodal sink (vii) Unstable defective node = defective nodal source (viii) Star (ix) Center (x)
View Full Document
Unlocking...