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1、四川师范大学 2009 至 2010 学年度第一学期常微分方程期末考试试题B 四川师范大学数软学院数学与应用数学期末考试 20082009 学年度第一学期期末考试 TEST PAPER OF ORDINARY DIFFERENTIAL EQUATIONS (B) 答卷说明：本试卷共6 个大题，满分 100 分，120 分钟完卷。 Questio n number I II III IV V VI total signatu re scores scores signature I (15 points). Solve the first order implicit differential
2、equation . scores signature II (15 points) Solve the linear differential equations . scores signature III (15 points).Solve the differential equation . scores signature IV(15 points) Solve the linear differential equation with the initial-value . scores signature V(20 points) If and , discuss of the
3、 and the stability of the trivial solution of the system . scores signature VI(20 points) Prove that is a general solution of the linear differential equation of first order . 常微分方程试题B参考答案及评分细则 I Solution . Let then the ordinary differential equation is rewrited , (1) Differentiate the above equatio
4、n with respect to x，then . 5 points It follows that . 10 points From (1), we have or The solutions of the ordinary differential equation are or . 15 points II Solution . Consider the coefficients matrix. The characteristic equation of A is det （）=0. =. The matrix A has three eigenvalues are . 5 poin
5、ts When ，from , we find . When ，from ，we find . . The fundamental matrix solution of is . 10 points Then, . The particular solution of the equation is =. The solution of linear ordinary differential equations is . 15points III Solution . Let P=2xy,Q= 2 y-3 2 x, then . 5 points The equation has integ
6、rating factor . Multiply the above equation by , then . 10 points Integrate the above equality , it follows that . 15 points IVSolution. The characteristic equation of the linear differential equation is The eigenvalues are （multiple number is three）. 5 points The fundamental solutions are . The sol
7、ution of the linear ordinary differential equation is 10 points Note that the initial-value conditions . Then, The solution of the initial-value problem is . 15 points V.Solution. Obviously, O(0,0) is a unique singular points. The system is linearized . (1) The characteristic equation is =, The eige
8、nvalues are . 6points 1) If , we have are positive ，real , and .Then the point (0,0) is an unstable improper node. 2) If , we have are conjugative imaginary numbers , and real of them are positive . Then the point (0,0) is an unstable spiral point. 3) If , we have are negative real , and .Then the p
9、oint (0,0) is an stable improper node. 4) If , we have are conjugative imaginary numbers , and real of them are negative . Then the point (0,0) is an stable spiral point. 14points Then, the trivial solution of system (1) is stable if and the trivial solution of system (1) is unstable if . We have th
10、at the trivial solution of the original system is stable if and the trivial solution of the original system is unstable if . 20 points VI. Proof. The associated homogeneous equation of the equation is , which has the solution . We assume is the general solution of nonhomogeneous equation ，then . 10p
11、oints Integrate the above equality，then . The general solution of non-homogeneous equation . 20 points 四川师范大学 2009 至 2010 学年度第一学期常微分方程期末考试试题B 四川师范大学数软学院数学与应用数学期末考试 20082009 学年度第一学期期末考试 TEST PAPER OF ORDINARY DIFFERENTIAL EQUATIONS (B) 答卷说明：本试卷共6 个大题，满分 100 分，120 分钟完卷。 Questio n number I II III IV V
12、VI total signatu re scores scores signature I (15 points). Solve the first order implicit differential equation . scores signature II (15 points) Solve the linear differential equations . scores signature III (15 points).Solve the differential equation . scores signature IV(15 points) Solve the line
13、ar differential equation with the initial-value . scores signature V(20 points) If and , discuss of the and the stability of the trivial solution of the system . scores signature VI(20 points) Prove that is a general solution of the linear differential equation of first order . 常微分方程试题B参考答案及评分细则 I S
14、olution . Let then the ordinary differential equation is rewrited , (1) Differentiate the above equation with respect to x，then . 5 points It follows that . 10 points From (1), we have or The solutions of the ordinary differential equation are or . 15 points II Solution . Consider the coefficients m
15、atrix. The characteristic equation of A is det （）=0. =. The matrix A has three eigenvalues are . 5 points When ，from , we find . When ，from ，we find . . The fundamental matrix solution of is . 10 points Then, . The particular solution of the equation is =. The solution of linear ordinary differentia
16、l equations is . 15points III Solution . Let P=2xy,Q= 2 y-3 2 x, then . 5 points The equation has integrating factor . Multiply the above equation by , then . 10 points Integrate the above equality , it follows that . 15 points IVSolution. The characteristic equation of the linear differential equat
17、ion is The eigenvalues are （multiple number is three）. 5 points The fundamental solutions are . The solution of the linear ordinary differential equation is 10 points Note that the initial-value conditions . Then, The solution of the initial-value problem is . 15 points V.Solution. Obviously, O(0,0)
18、 is a unique singular points. The system is linearized . (1) The characteristic equation is =, The eigenvalues are . 6points 1) If , we have are positive ，real , and .Then the point (0,0) is an unstable improper node. 2) If , we have are conjugative imaginary numbers , and real of them are positive
19、. Then the point (0,0) is an unstable spiral point. 3) If , we have are negative real , and .Then the point (0,0) is an stable improper node. 4) If , we have are conjugative imaginary numbers , and real of them are negative . Then the point (0,0) is an stable spiral point. 14points Then, the trivial
20、 solution of system (1) is stable if and the trivial solution of system (1) is unstable if . We have that the trivial solution of the original system is stable if and the trivial solution of the original system is unstable if . 20 points VI. Proof. The associated homogeneous equation of the equation is , which has the solution . We assume is the general solution of nonhomogeneous equation ，then . 10points Integrate the above equality，then . The general solution of non-homogeneous equation . 20 points