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︰夏俊雄 開課學院：理學院 開課系所︰數學系 考試日期︰2020年11月17日(二)，15:30-18:30 考試時限：180分鐘 試題 : 　　　　　　　　　　　　　　ODE EXAM 2 11/17/2020 1. (20 points) Set (3 1 -2) 　　　　　　　　　　　　　　　A = (2 4 -4), (2 1 -1) T X(t) = (x1(t),x2(t),x3(t)) and g(t) = (t,cos(t),sin(t)). Solve the differential system 　　　　　　　　　　　　　X'(t) = AX(t) + g(t) T with initial condition X(0) = (1,2,3) . 2. (10 points) Find the solution of the initial value problem 2y'' + y' + 3y = δ(t-5) with initial condition y(0) = 0, y'(0) = 0. 3. (20 points) Solve the following two integro-differential equations 1 t 2 ψ'(t) - --- ∫(t-ξ) ψ(ξ)dξ = -t, ψ(0) = 1, 2 0 t ψ'(t) + ψ(t) = ∫sin(t-ξ)ψ(ξ)dξ,　　ψ(0) = 1. 0 4. Consider the differential equation 2 x'''(t) + t x''(t) + (sin(t))x'(t) - (cos(t))x(t) = 0. (0.1) Suppose that x1(t), x2(t) and x3(t) are three solutions to (0.1) with initial conditions x1(0) = 1, x1'(0) = 0, x1''(0) = 0, x2(0) = 0, x2'(0) = 1, x2''(0) = 0, x3(0) = 0, x3'(0) = 0, x3''(0) = 1. (i) (10 points) Calculate the explicit form of the Wronskian function W[x1(t),x2(t),x3(t)]. (ii) (20 points) Show that for any other solution y(t) of (0.1), there exists unique constants a, b, c such that y(t) = ax1(t) + bx2(t) + cx3(t). (Note: To get full points, we ask you to express a, b, c in terms of y(0), y'(0) and y''(0) and prove ax1(t) + bx2(t) + cx3(t) equals y(t) for all t∈|R by Wronskain method.) 5. (10 points) Let x(t) be the solution of the differential equation -t x'(t) + (2+sin(t))x(t) = e sin(t) (0.2) with initial condition x(0) = 1. Find lim x(t). t→∞ 6. (10 points) Solve the differential equation 2 t y''(t) + ty'(t) - y(t) = 0 for t≧1 with y(1) = 2 and y'(1) = 0. --

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