課程名稱︰常微分方程導論
課程性質︰數學系大二必修
課程教師︰林紹雄
開課學院：數學系
開課系所︰數學系
考試日期（年月日）︰104年01月10日
考試時限（分鐘）：180分鐘
試題 :
The following problems has a total of 140 points. Please write down your
computational steps, or proof reasoning clearly on the answer sheets.
A.Apply the method of the Laplace transforms to solve the following problems.
Each has 11 points.
(a)Solve the system x"+x-y=0, y"+y-x=tδ(t-2), x(0)=y(0)=0, x'(0)=-2, y'(0)=1
(b)Find L{J1(t)}, where J1(t) is the Bessel function of order 1
(c)Solve y'+y-∫y(η)sin(t-η)dη(from 0 to t)=-cost, y(0)=1
B.(20 points) Solve xy"+y'+y=0 by the power series method in x>0. What are the
region of convergence for the solutions?
C.(16 points) Solve the Riccati equation IVP problem y'=y^2+x^3, y(0)=0
D.(18 points) Find the equilibria of the plane system x'=y-x^2+2, y'=x^2-xy.
Draw the phase portrait of the linearized plane system at each equilibrium,
and determine its type.
E.(20 points) Draw the phase portrait of the prey-predator system x'=x(2-ax-y)
, y'=y(-1+ax) (where a>0 is a constant) in x≧0, y≧0. Discuss the type of
each equilibria first, then apply the Poincare-Bendixon theorem to finish the
draw. What is ω((α,β)) for all α≧0, β≧0 ?
F.Let f: R → R be a C1 function. The phase coordinate (x,y) for the equation
x"+f(x)=0 is given by y=x'
(a)(9 points) Prove that every equilibria are of the form (x*,0) with
f(x*)=0, and when f'(x*)≠0, this equilibrium is either a center or a saddle.
(b)(8 points) Prove that this system has no limit cycles.
G.Determine which of the following statements is true. Prove your answer. Each
has 8 points.
(a)x=0 is an irregular singular point of x^3*y"-xy'+y=0. Hence it has no
∞
solution of the form y=x^r*Σ anx^n in x>0.
n=0
(b)If p∈R^2 is a stable center for the linearized plane system of x'=f(x)
(where x∈R^2, and f(p)=0), then p is also a stable equilibrium of x'=f(x).