私立中原大學八十九學年度博士班招生考試命題紙

所組別：

數學系

科目：

微分方程

考試時間：

6月14日第2節

1.	(20 points)Consider the initial value problem 　　x'=f(t,x), x(t 0 )=x0 with , a and d are positive constants. Assume that f(t,x) is continuous in G=[t0-a,t0+a]×D and f(t,x) is Lipschitz in x. Show that the initial value problem gas one and only one solution for |t-t0|≦inf(a,d/M) with
2.	(5 points) State the Poincar'e-Bendixson theorem.
3.	(15 points) Consider the equation 　　 Assume that p(x) and q(x) are smooth and p(x)>0 for all x. Show that above equation has no periodic solutions.
4.	(15 points) Determine the stability of the solution (x,y)=(0,0) of the system 　　x'=2xy+x3 　　y'=x2-y5.
5.	(15 points) Find the regions in the xy plane where the equation 　　(1+x)u xx +2xyu xy -y2u yy =0 is elliptic, hyperbolic and parabolic.
6.	(15 points) Solve the initial value problem uy=xuux,　u(x,0)=x.
7.	(15 points) Let be a solution of Here Ω is a bounded region in Rn with smooth boundary. Coefficient functions ak(x) and c(X) are smooth with c(x)<0 in Ω. Show that u=0 on implies u=0 in Ω.

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