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. --

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