课程名称︰常微分方程导论 课程性质︰数学系大二必修 课程教师︰林绍雄 开课学院：理学院 开课系所︰数学系 考试日期︰2011年01月08日(六)，09:00-12:00 考试时限：180分钟 是否需发放奖励金：是 试题 : 　　　　　　　　　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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