推 ericswatch :好快 11/06 13:23
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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