课程名称︰常微分方程导论 课程性质︰数学系大二必修课 课程教师︰夏俊雄 开课学院：理学院 开课系所︰数学系 考试日期︰2020年10月13日(二)，15:30-18:30 考试时限：180分钟 试题 : 　　　　　　　　　　　　　　ODE EXAM 1 10/13/2020 1. Set 　　　　　　　　　　　　　　A = ( 0 1), (0.1) (-1/4 1) T and X(t) = (x1(t),x2(t)). tA t^k k (a) (15 points) Calculate e := I + tA + … + ----- A + …. 　　　　　　　　　　　　　　　　　　　　　　　　　　k! 　 (b) (15 points) Solve the differential system 　　　　　　　　　　　　x1'(t) = x2(t) + e^t, 1 (0.2) x2'(t) = - --- x1(t) + x2(t) + e^(2t), 4 T with the inital condition X(0) = (1,2) . 2. (20 points) Solve the differential equation x'''(t) + x''(t) - x'(t) - x(t) = 0, with the initial condition x(0) = x'(0) = 0, x''(0) = 1. 3. (15 points) Suppose that f(t) and g(t) are solutions of the differential equation 2 f'(t) + tf(t) = t . (0.3) Show that (a) If f(0) = 1, then f(t) > 0 for all t > 0. (b) If g(0) > g(0), then f(t) > g(t) for all t > 0. (c) If f(0) < 0, then there exists exactly one moment t = t0 > 0 such that f(t0) = 0. 4. (20 points) Solve the differential equation x'(t) + (sin(t))x(t) = sin(t) (0.4) with initial condition x(0) = 0. 5. (15 points) Let y1(t) and y2(t) are two solutions of the differential equation y''(t) + ty'(t) + q(t)y(t) = 0, with initial conditions y1(0) = 1 = y2'(0), y1'(0) = 0 = y2(0), where q1(t) is a smooth function that we do not have exact information. Calculate det(y1(100) y2(100) ). (0.5) (y1'(100) y2'(100)) --

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