課程名稱︰常微分方程導論
課程性質︰必修
課程教師︰夏俊雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2012/11/30
考試時限（分鐘）：100分鐘
試題 :
ODE Midterm Exam 11/30/2012
1. (10 points) Solve the differential equation:
dy 6x^5 - 2x + 1
── = ─────── (0.1)
dx cosy + e^y
near (x,y)=(0,0).(For the solution,an implicit function of x and y is expected)
2. (20 points) Find the solution to
1 dy 2y
── - ── = xcosx, x>0, (0.2)
x dx x^2
with the initial condition y(π) = π^2.
3. (20 points) Find a time periodic solution for the system of differential
equations:
╭ dx
｜── = x + 3y + sint,
╱ dt (0.3)
╲ dy
｜── = 3x + y - cost.
╰ dt
4. (30 points) Find the general solutions of the following differential
equations and classify the maximal intervals of existence (which might depend
on the choices of initial conditions).
(a) x'' - 2x' + x = (t^2 + t)e^t + 2t + 1,
(b) x' = 3(3 - x)(x + 1).
5. (20 points) Calculate the inverse Laplace tranbsforms of the following
functions.
s - 1 3
F(s) = ──────, F(s) = ──────.
s^2 - 2s + 5 (s^2 + 9)^2
6. (10 points) Suppose F(x) is a continuous and bounded function defined on R
and the differential equation
x'(t) = F(x), (0.4)
always has a unique solution. We denote the solution to the above equation with
initial condition x = a by x(t) = ψ(t,a). Prove that
lim ψ(t,y) → ψ(t,a). (0.5)
y→a