課程名稱︰常微分方程導論
課程性質︰數學系大二必修課
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10月13日(二)，15:30-18:30
考試時限：180分鐘
試題 :
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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