課程名稱︰常微分方程導論 課程性質︰系必修 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（年月日）︰2010年11月06日 考試時限（分鐘）：180分鐘 是否需發放獎勵金：是 試題 : There are problems A to F with a total of 120 points. Please write down your compitational or proof clearly on the answer sheets. A. Solve the following first-order ODE. You can express the solutions in expl- icit functions, or in implicit functions. Each has 8 points. (a) (x + y - 4)dy + (x - 3y -4)dx = 0 , y(4) = 2. 3 + y^2 +xy (b) y' = － -------------. 3 + x^2 +xy (c) (x^2 ×y^3 + y)dx + xdy = 0 , y(-1) = -2. B.(16 points) It is known that the homogeneous part of the ODE (1 - t^2)y'' -2ty' +6y = t has a a polynomaial solution. Use this fact to find the general solutions of the equation in |t| < 1. C.(15 points) Solve (x^3)y'''+ (2x^2)y''-xy' +y = x^3 -x(ln |x|)^2 + (ln |x|)/x in x < 0. D. Let A be the 4 ×4 matrix [ 3 -4 1 0 ] A = [ 4 3 0 1 ]. [ 0 0 3 -4 ] [ 0 0 4 3 ] (a)(10 points) Find a basis for the solution space of the homogeneous equation y' = Ay. (b)(5 points) Compute e^(tA) explicitly. (c)(10 points) Apply the method of undetermined coefficeients to find a parti- cular solution of the inhomogeneous equation y' = Ay + tωe^(3t)(sin 4t), where ω = [0 0 0 1]^T. E. (12 points) Let A(t) be continuous n ×n matrix-valued function defined in t ∈ R such that A(t + T) = A(t) for all t ∈ R, where T > 0 is a constant. If Φ(t) is a fundamental matrix for the linear system y' = A(t)y, prove that there exists a constant matrix C such that Φ(t + T) = Φ(t)C for all t ∈ R. F. Determine which of the following statements is true. Prove your answer, or give a counterexample. Each has 7 points. (a) The IVP y' = y sin(1/y) , y(0) = 0 has an unique solution which is 0. Note that the function y sin(1/y) is defined to be 0 at y = 0. (b) There exist two continuous functions p(t) and q(t) defined in the interval (-1, 1) such that y = sin(t^2) is a solution of the ODE y''+ p(t)y' +q(t)y = 0 in |t| < 1. (c) Let A, B be two n ×n matrices. Then AB = BA iff e^[t(A+B)] = e^(tA) e^(tB) for all |t| < 1. (d) Let be a n ×n matrix. Then every solution y(t) of the linear system y'=Ay is bouned as t → ∞ iff the real parts of the eigenvalues of A are not positive. --

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