課程名稱︰常微分方程導論
課程性質︰必修
課程教師︰林紹雄
開課學院：數學系
開課系所︰數學系
考試日期（年月日）︰103年11月08日
考試時限（分鐘）：180分鐘
試題 :
A.Solve the following ODEs. You may express the solutions in explict functions
, or in integral forms. Each has 10 points.
(a)y'=y(y^3-x), y(0)=4
(b)y(x+y+1)dx+(x+2y)dy=0, y(-1)=1
(c)xy'=yln(xy)
(d)y'''-2y''+y'=xe^x+5, y(0)=2, y'(0)=2, y''(0)=-1
B.It is known that the homogeneous part of the ODE ty''-(1+t)y'+y=(t^2)(e^(2t))
has a polynomial solution. Use this fact to find the general solution of this
ODE in t > 0.
C.Let A = [ 4 0 1]
[ 0 6 0]
[-4 0 4]
(a)Find a basis for the solution space of the homogeneous equation y'=Ay
(b)Use (a) to write down e^(tA)
(c)Solve the inhomogeneous equation y'=Ay+(te^(4t)sin2t)w either by the
method of underdetermined coefficients, or by the method of variations,
where: T
w = [0 0 2]
D.Let p(t) and q(t) be continuous in I=(a,b). y1(t) and y2(t) are a basis of
the solution space of y''+p(t)y'+q(t)y=0. Prove the following statements.
(a)If y1(t0) = 0 at some t0∈I, prove that y2(t0)≠0, and ther exists h > 0
such that y1(t)≠0 for all t∈I with 0 < |t-t0| < h.
(b)If y1(t0) = y1(t1) = 0 for some to, t1∈I with t0 < t1, prove that there
exists some with t0 < t2 < t1 such that y2(t2) = 0
E.Determine which of the following statements is true. Give sufficient
reasoning to support your answer. Each has 6 points.
(a)There exist two continuous functions p(t) and q(t) in the interval
I=(-1,1) such that y(x) = sin(t^2) is a solution of the ODE y''+p(t)y'
+q(t)y = 0 for t∈I
(b)Let f:R→R be continuously differentiable, and f(y)≠0 for all y. Then
lim |y(t)|=∞ for all solutions of the equation y'=f(y)
t→∞
(c)Since y^(1/3) is not a Lipschitz continuous function near y = 0, a
solution y(t) of y'=y^(1/3) which is defined for all t∈R cannot be
uniquely determined by the condition y(1) = 1
(d)Let A∈M(2). Then all solutions of the linear system y'= Ay are bounded
as t→∞ iff the real parts of the eigenvalues of A are not positive.
(e)Let A,B ∈ M(n). Then e^(t(A+B))=(e^tA)(e^tB) for all t iff AB = BA.