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︰2011年01月08日(六)，09:00-12:00
考試時限：180分鐘
是否需發放獎勵金：是
試題 :
Math 201-24900 (ODE) Final (01/08/2010)
There are problems A to G with a total of 140 points. Please write down your
computational or proof steps clearly on the answer sheets.
A. Apply the method of the Laplace transforms to solve the following problems.
Each has 10 points.
(a) Solve the system x''+ x' + y' + 2x - y = 0, y''+ x' + y' + 4x - 2y = 0;
x(0) = y(0) = 1, x'(0) = y'(0) = 0.
(b) Find the Laplace transform of (J_0)(2√t), where (J_0)(x) is the Bessel
function of order 0.
2 2
(c) Solve the ODE a y'' - b y = (-1/2)δ(t) for all t∈|R such that y(-t) =
y(t) for t∈|R and lim y(t) = 0, where a and b are constants.
t→∞
B. (18 points) Solve the linear equation xy''-y = 0 by the power series method.
2 2
C. (19 points) Find the solution of the IVP y' = x + y , y(0) = 0.
D. (15 points) Find all equilibria of the planar system x'= xy - 2, y'= x - 2y.
Draw the phase portrait of the linearized planar system at each equilibrium,
and determine its type.
E. (18 points) Draw the phase portrait of the prey-pradator system
dx/dt = 30x - 2x^2 - xy, dy/dt = 20y - 4y^2 + 2xy. Discuss the type of each
equilbria first. Do we have coexistence in this system?
F. (12 points) Consider the following one-step implicit scheme
h
y_(n+1) - y_n = ---[4f(x_n,y_n) +2f(x_(n+1),y_(n+1))+hg(x_n,y_n)] for n=1,3,.
6
with y_0 = η for solving the IVP dy/dx = f(x,y), y(a) = η, where h is the
step-size, x_n = a + nh, and f(x,y) is differentiable to any desired order.
Show that the scheme is consistent, and find its order. Determine the
conditions of absolute stability for the scheme when applied to f(x,y) = -λy
with λ > 0. g(x,y) = (f_x)(x,y) + (f_y)(x,y) * f(x,y)
G. Determine which of the following statements is true. Prove your answer, or
give a counterexample. Each has 7 points.
(a) The point x = 0 is an irregular signular point for the equation
(x^3)y'' - xy' + y = 0. Hence this equation has no Frobeneous solution
∞
of the form y(x) = (x^r)Σ (c_n)(x^n) in x > 0.
n=0
(b) The planar Hamiltonian system dx/dt = ∂H/∂y, dy/dt = -∂H/∂x (where
H(x,y) is C^2) has no limit cycle because it has not any cycle orbits.
(c) The initial vaule problem
(x^2)(x-2)e^[(2x-x^2) / (x-1)^2] y''+ (x^3+3x^2-10x)y' - (x-8)y = 0,
y(1) = 0, y'(1) = 1 has an unique strictly increasing solution defined
in 0 < x <2. The function e^[-(2x-x^2) / (x-1)^2] is defined to be 0
at x = 1.
(d) If (0,0) is a stable center point for the linearized system around (0,0)
of the planar system dx/dt = F(x,y), dy/dt = G(x,y) (where F(x,y) and
G(x,y) are C^1 functions), then (0,0) is also a stable equilibrium for
this planar system.
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