课程名称︰常微分方程导论
课程性质︰数学系大二必修
课程教师︰陈俊全
开课学院：理学院
开课系所︰数学系
考试日期︰2017年11月17日(五)
考试时限：08:10-10:00，计110分钟
试题 :
Choose 4 from the following 6 problems
1. (25%) Solve the following equations.
(a) dy/dx = (y sin(x) - y)/(y^2+1).
(b) 2(x^2-y^2)dx + xydy = 0, y(1) = 2.
(c) y sin(2x)dx + (sin(x))^2 dy = 0.
2. (25%) Solve the equations.
(a) y'' + 2y' + 7y = 0, y(0) = 1, y'(0) = -2.
(b) y'' - 2y' + y = e^t.
(c) y'' - 2y' + y = e^t/(t^2+1). (Hint: variation of parameters.)
3. (25%)
(a) Prove that the initial value problem
y'(t) = y /(1+t^2 + y^2), y(0) = 1.
has at most one solution. (Hint: Gronwall's inequality.)
(b) Show that the initial value problem
y' = y^(1/5), y(0) = 0
has more than one solution.
4. (25%) Suppose that the population y of a certain species of fish is
described by the Schaefer model with harvesting
dy/dt = r(1 - y/K)y - Ey, y(0) = Y_0,
where r, K and E are positive constants and E < r.
(a) Show that there are two equilibrium points y1 = 0 and y2 = K(1 - E/r).
(b) Show that if 0 < Y_0 < y2, then y(t) is increasing for t>0 and
lim y(t) = y2.
t->∞
(c) Show that y = y2 is asymptotical stable and y = y1 is ubstable.
5. (25%) Let t
ψ (t) = 0 , ψ (t) = ∫sψ (s)ds + 1, n = 0,1,2,3,...
0 n+1 0 n
(a) Show that 0≦ψ (t)≦2 for 0≦t≦1.
n
(b) Show that lim ψ (t) exists for 0≦t≦1.
n→∞ n
(c) Find lim ψ (t) for 0≦t≦1.
n→∞ n
6. (25%) A swimming pool whose volume is 10,000 gal contains water that is
0.02% chlorine. Starting at t = 0, city water containing 0.001% chlorine
is pumped into the pool at a rate of 5 gal/min. The pool water flows out
at the same rate. What is the percentage of chlorine in the pool after 1
hour? When will the pool water be 0.004% chlorine?