Be sure that this examination has three pages.
The University of British Columbia
Final Examinations – December 2006
Mathematics 215
Elementary Differential Equations I
Mathematics 255
Ordinary Differential Equations
Time: 21
2hours
Special instructions:
(1) One 81
2
00 ×1100 page of notes may be used, but no other aids are permitted. In particular,
calculators and cell phones are not allowed.
(2) Answers must be justified to receive full credit.
(3) A table of Laplace transforms is attached.
Marks 1. (a) Find the general solution of the equation
[15] dy
dt =t2−y
t.
(b) Find all values of the constants aand bso that the differential equation
dy
dx =−ya
2xy + 2bxy2
is exact, then for these values of aand bfind the solution of the equation that
also satisfies the initial condition y(1) = 1.
2. Consider the autonomous equation
[20] dy
dt = (y−1)(y−5),−∞ < y < +∞.
(a) Find all critical points (equilibria) and sketch the one-dimensional phase portrait
(i.e. phase line) for the equation.
(b) If y1(t) and y2(t) are two solutions of the equation which satisfy the initial condi-
tions y1(0) = 4 and y2(0) = 3, find limt→∞ |y1(t)−y2(t)|if it exists.
(c) Plot the direction field of the equation, and on the same plot sketch the graph
of the solution y1(t) versus t, if y1(t) solves the initial value problem with initial
condition y1(0) = 4. You do not need to find the solution y1(t) explicitly, but
indicate clearly where the solution is increasing or decreasing, and where its graph
is concave up or concave down.
