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）︰2012/12/14
考试时限（分钟）：
试题 :
ODE QUIZ 7 12/14/2012
You need to turn in Problems 1,2,4,5 in class. In this quiz, let f(t) and g(t)
be piecewise continuous finction 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.Calculate sint*sint.
1
2.Calculate the inverse Laplace transform of ──────.
(s^2 + 1)^2
3. 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),
and
L^-1(L(f)(s)L(g)(s)) = (f*g)(s).
4. Solve the integro-differential equation
y'(t) = 1 - ∫y(t-s)(e^-2s)ds, y(0) = 1.
5. Do you think the following equations have periodic solutions? Prove or
disprove it.
x'' - x + x^3 = 0,
x'' + x^2 - x^4 = 0.