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：理學院
開課系所︰數學系
考試日期︰2010年12月14日(二)，14:30-17:20
考試時限：50分鐘
是否需發放獎勵金：是
試題 :
Math 201-24900 (ODE) Quiz No.6 (12/14/2010)
There are problems (A) to (D) with a total of 50 points. Please write down your
computational or proof steps clearly on the answer sheets.
Consider the following plane autonomous system
dx 2 dy
---- = -3x - 4y - 1, ---- = 4y - 4x.
dt dt
A. (10 points) Find its equilibra, and determine their stability.
B. (20 points) Draw the phase portraits of the linearized systems around each
equilibrium.
3 2
C. (10 points) Define the function G(x,y) = x + 4xy - 2y + x. If (x(t),y(t))^T
is an orbit of the system, prove that d/dt (G(x(t),y(t))≦0 for all t, and
if d/dt (G(x(t),y(t)) = 0 at some t iff this orbit is an equilibrium.
D. (10 points) Use (c) to probe that the systme has no cycle orbit.
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