课程名称︰常微分方程导论
课程性质︰必修
课程教师︰夏俊雄
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2012/12/28
考试时限（分钟）：100分钟
试题 :
ODE ExAm 3 12/28/2012
In this exam, let f(t) and g(t) be piecewise continuous function defined on
[0,∞). The convolution of f(t) and g(t) are defined as
t
(f*g)(t) := ∫ f(t-s)g(s)ds.
0
1. (10 points) Let L denote the Laplace transform and L^-1 denote the inverse
Laplace transform. Show that
L(f*g)(s) = L(f)(s)L(g)(s).
2. (10 points) Solve the integro-differential equation
t
y'(t) = 1 - ∫ y(t - s)e^-3s ds, y(0) = 2.
0
3. (10 points) Do you think the following equation have periodic solutions?
Prove or disprove it.
x'' + x + x^3 = 0.
4. (80 points) Find general solution for the following differential equations.
(1) (t^2)x'' - 3tx' + 4x = 0, t>0,
(2) x'''(t) - 3x''(t) + 3x'(t) - x(t) = 2t + e^t, t∈R,
(3) x''(t) + tx'(t) + 2x(t) = 0, t∈R,
(4) y'' + y = u_4π(t), t≧0.
5. (10 points) Show that for certain μ∈R, the van der Pol equation
u'' - μ(1 - u^2)u' + u = 0
has time periodic solutions. What theorem shall you apply?