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2012常微分方程试题B及答案.pdf

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1、第 1 页共 4 页南京农业大学试题纸2011-2012学年第 2 学期课程类型：必修试卷类型：B 课程常微分方程班级学号姓名题号一二三四五六七八九总分签名得分一、填空题（每小题3 分，本题共 30 分）1方程2231)(dsrddsdr是阶方程2.若y=y1(x)，y=y2(x)是一阶线性非齐次方程的两个不同解，则用这两个解可把其通解表示为_.3若)(),.(),(21txtxtxn为 n 阶齐次线性方程在区间I 上的n 个解，则它们线性无关的充要条件是_.4.方程0),(),(dyyxNdxyxM有只含x的积分因子的充要条件是.5.方程21ddyxy的常数解是6n阶线性

2、齐次微分方程的所有解构成一个维线性空间7.李普希兹条件是保证一阶微分方程初值问题解惟一的条件8.求(,)dyf x ydx满足00)(yx的解等价于求积分方程_的连续解.9.函数()atf te的 Laplace 变换是10.方程212ydxdy经过（0，0）点的解的存在区间是二、计算题（每小题5 分，本题共 20 分）求解下列微分方程：11.xyxy2e3dd本试卷适应范围信息与计算科学装订线装订线第 2 页共 4 页12.0 xxx130)2()(2dyyxdxyx14.52dyxydxxy三、计算题（每小题10 分，本题共 30 分）15.求下列方程组的通解yxtyyxtx4dddd文

3、档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2

4、Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10

5、I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J

6、4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3

7、L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y

8、10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L

9、1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4第 3 页共 4 页16.用常数变易法求解一阶非齐次线性微分方程()().dyP x yQ xdx17.讨论方程23dxdy31y在怎样的区域中满足解的存在唯一性定理的条件，并求通过点（0，0）的一切解.四、证明题（每小题10 分，本题共 20 分）18.设),(

10、yxf及yf连续,试证方程0),(dxyxfdy为线性方程的充要条件是它有仅依赖于x的积分因子.文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:C

11、A9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS

12、2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR

13、3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码

14、:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4

15、HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2

16、ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4第 4 页共 4 页19设(),()p x q x都是区间(,)上的连续函数,且(),()xx是二阶线性方程0)()(yxqyxpy的一个基本解组.试证明:(i)()

17、,()xx都只能有简单零点(即函数值与导函数值不能在一点同时为零)；(ii)(),()xx没有共同的零点；(iii)(),()xx没有共同的零点.文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8

18、X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3

19、U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA

20、9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2

21、R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3

22、I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:

23、CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4第 5 页共 4 页常微分方程模拟试题(B)参考答案2012.7一、填空题（每小题3 分，本题共30 分）1二2)(

24、)()(1211xyxyxyC3()0W t或00()=0,W ttI4)(xNxNyM5.1y6.n7.充分8.00(,)xxyyf x y dx9.1,Resasa10.,二、计算题（每小题5 分，本题共20 分）11.解：齐次方程的通解为xCy3e(3 分)令非齐次方程的特解为xxCy3e)(代入原方程，确定出CxCx5e51)(原方程的通解为xCy3e+x2e51（5 分）12.解：对应的特征方程为：012，解得ii23,23212211（3 分）所以方程的通解为：)23sin23cos(2121tctcext（5 分）13.1yM，xN=1,xNyM所以此方程是恰当方程.（3 分）凑

25、微分，0)(22xdyydxydydxx得Cyxyx2331（5 分）145,1,dydtxytdxdx令则1,(7)77dtttdtdxdxt原方程化为：变量分离（3 分）21772txct两边积分217(5)7.2(5)xyxcxy代回变量（5分）文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8

26、Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X

27、5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U

28、4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9

29、V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R

30、8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I

31、3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4第 6 页

32、共 4 页三、计算题（每小题10 分，本题共30 分）15特征方程为01411EA,即0322.特征根为31，12.（4 分）31对应特征向量应满足0031413111ba可确定出2111ba同样可算出12对应的特征向量为2122ba所以，原方程组的通解为ttttCCyx2ee2ee2331（10 分）.16解：(),dyP x ydx（1）这是一个变量分离方程，通解为(),P x dxyce这里 c 是任意常数。（4 分）假设()()P x dxyc x e是()()dyP x yQ xdx的通解，代入方程，则有()()()P x dxdc xQ x edx积分后得到()()(),P x d

33、xc xQ x edxc（8 分）这里c是任意常数，方程的通解为()()()P x dxP x dxyeQ x edxc（10 分）17.解：设 f(x,y)=2331y,则)0(2132yyyf文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L

34、1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8

35、Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X

36、5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U

37、4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9

38、V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R

39、8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4第 7 页共 4 页故在0y的任何区域上yf存在且连续，因而方程在

40、这样的区域中满足解的存在唯一性定理的条件.（4 分）显然，0y是通过点（0，0）的一个解；（6 分）又由23dxdy31y解得，|y|=23)(cx所以，通过点（0，0）的一切解为0y及|y|=是常数0),()()(023ccxcxcx（10 分）四、证明题（每小题10 分，本题共20 分）18.证明：必要性若该方程为线性方程,则有)()(xQyxPdxdy,此方程有积分因子dxxPex)()(,)(x只与x有关 .（4 分）充分性若该方程有只与x有关的积分因子)(x,则0),()()(dxyxfxdyx为恰当方程,从而dxxdyyxfx)(),()(,)()(xxyf,)()()()()()

41、()()(xQyxPxQyxxxQdyxxf.其中)()()(xxxP.（8 分）于是方程可化为0)()(dxxQyxPdy即方程为一阶线性方程.（10 分）19证明：(),()xx的朗斯基行列式为()()()()()xxW xxx因(),()xx是基本解组,故文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:

42、CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 H

43、S2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 Z

44、R3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编

45、码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4

46、 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2

47、 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4第

48、 8 页共 4 页()0,()W xxR.若存在0 xR,使得00()()0 xx,则由行列式性质可得0()0W x,矛盾.即()x最多只能有简单零点.同理对()x有同样的性质,故(i)得证.若存在0 xR,使得00()()0 xx,则由行列式性质可得0()0W x,矛盾.即(),()xx无共同零点.故(ii)得证若存在0 xR,使得00()()0 xx则同样由行列式性质可得0()0W x,矛盾.(),()xx没有共同的零点.故(iii)得证.文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I

49、3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:C

50、A9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS2R8X5Y10I2 ZR3I3U4L1J4文档编码:CA9V8Q3L2Y4 HS

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