課程名稱︰微積分乙下
課程性質︰大一共同必修
課程教師：陳其誠
開課系所︰醫學院、公衛學院等
考試時間︰2006.3.28
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculations and reasoning.
1. Solve the following differential equations (6 points each):
(a) dy/dx = e^y, where y(0) = 4 (b) dx/dt = t/(1+t^2), where x(0) = 2
(c) dx/dt = 0.1(1-x),where x(0) = 0.5 (d) dp/dt = p(1-p), where p(0) = 0.5
(e) dp/dt = p(1-p), where p(0) = 1
2. Suppose the function N(t) satisfies dN/dt = 100-N. Find all the possible
values of lim N(t). (10 points)
t→∞
3. Suppose that dy/dt = y(1-y)(3-y)
(a) Find the equilibria of this differential equation. (6 points)
(b) Suppose that y(0) = 2. Show that for t≧0, y(t) is a decreasing function
(10 points)
(c) Discuss the stability of each equilibrium. (9 points)
4. Consider the Predator-Prey Equations
db/dt = (1 - 0.01p)b dp/dt = (-1 + 0.01b)p
(a) Find every equilibrium of the system (7 points)
(b) Show that if 0≦b(t0)≦100 and 0≦p(t0)≦100 then at t0, b is increasing
and p is decreasing (8 points)
5. Consider the Competition Equations
da/dt = a(1 - 0.1a - 0.3b) db/dt = b(1 - 0.3a - 0.1b)
(a) Find every equilibrium of the system (8 points)
(b) Show that two of the equilibria are stable and the others are unstable
(8 points)
(c) Show that if a(0) = b(0) then a(t) = b(t) for every t (4 points)
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