[試題] 109上 夏俊雄 常微分方程導論 第二次期中
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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