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1、1 第七章常微分方程一变量可分离方程及其推广1变量可分离的方程（1）方程形式：0yQyQxPdxdy通解CdxxPyQdy（注：在微分方程求解中，习惯地把不定积分只求出它的一个原函数，而任意常数另外再加）（2）方程形式：02211dyyNxMdxyNxM通解CdyyNyNdxxMxM12210,012yNxM2变量可分离方程的推广形式（1）齐次方程xyfdxdy令uxy，则ufdxduxudxdycxcxdxuufdu|ln二一阶线性方程及其推广1一阶线性齐次方程0yxPdxdy它也是变量可分离方程，通解dxxPCey，（c为任意常数）2一阶线性非齐次方程xQyxPdxdy用常数变易法可求出通
2、解公式令dxxPexCy代入方程求出xC则得CdxexQeydxxPdxxP3伯努利方程1,0yxQyxPdxdy令1yz把原方程化为xQzxPdxdz11再按照一阶线性非齐次方程求解。4方程：xyPyQdxdy1可化为yQxyPdydx以y为自变量，x为未知函数再按照一阶线性非齐次方程求解。三、可降阶的高阶微分方程方程类型解法及解的表达式xfyn通解nnnnnnCxCxCxCdxxfy12211次yxfy,令py，则py，原方程pxfp,一阶方程，设其解为1,Cxgp，即1,Cxgy，则原方程的通解为21,CdxCxgy。yyfy,令py，把p看作y的函数，则dydppdxdydydpdxd
3、py把y，y的表达式代入原方程，得pyfpdydp,1一阶方程，设其解为,1Cygp即1,Cygdxdy，则原方程的通解为21,CxCygdy。2 四线性微分方程解的性质与结构我们讨论二阶线性微分方程解的性质与结构，其结论很容易地推广到更高阶的线性微分方程。二阶齐次线性方程0yxqyxpy（1）二阶非齐次线性方程xfyxqyxpy（2）1若xy1，xy2为二阶齐次线性方程的两个特解，则它们的线性组合xyCxyC2211（1C，2C为任意常数）仍为同方程的解，特别地，当xyxy21（为常数），也即xy1与xy2线性无关时，则方程的通解为xyCxyCy22112若xy1，xy2为二阶非齐次线性方程
4、的两个特解，则xyxy21为对应的二阶齐次线性方程的一个特解。3若xy为二阶非齐次线性方程的一个特解，而xy为对应的二阶齐次线性方程的任意特解，则xyxy为此二阶非齐次线性方程的一个特解。4若y为二阶非齐次线性方程的一个特解，而xyCxyC2211为对应的二阶齐次线性方程的通解（1C，2C为独立的任意常数）则xyCxyCxyy2211是此二阶非齐次线性方程的通解。5设xy1与xy2分别是xfyxqyxpy1与xfyxqyxpy2的特解，则xyxy21是xfxfyxqyxpy21的特解。五二阶和某些高阶常系数齐次线性方程1二阶常系数齐次线性方程0qyypy其中p，q为常数，特征方程02qp特征方
5、程根的三种不同情形对应方程通解的三种形式（1）特征方程有两个不同的实根1，2则方程的通解为xxeCeCy2121（2）特征方程有二重根21则方程的通解为xexCCy121（3）特征方程有共轭复根i，则方程的通解为xCxCeyxsincos212n阶常系数齐次线性方程012211ypypypypynnnnn其中nipi,2,1为常数。相应的特征方程012211nnnnnpppp特征根与方程通解的关系同二阶情形很类似。（1）若特征方程有n个不同的实根n,21则方程通解xnxxneCeCeCy2121（2）若0为特征方程的k重实根nk则方程通解中含有y=xkkexCxCC0121（3）若i为特征方程
6、的k重共轭复根nk2，则方程通解中含有xxDxDDxxCxCCekkkkxsincos121121由此可见，常系数齐次线性方程的通解完全被其特征方程的根所决定，但是三次及三次以上代数方程的根不一定容易求得，因此只能讨论某些容易求特征方程的根所对应的高阶常系数齐次线性方程的通解。文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F1
7、0O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 H
8、T1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S1
9、0J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:
10、CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M
11、1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8
12、ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3
13、R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X63 六、二阶常系数非齐次线性方程方程：xfqyypy其中qp,为常数通解：xyCxyCyy2211其中xyCxyC2211为对应二阶常系数齐次线性方程的通解上面已经讨论。所以关键要讨论二阶常系数非齐次线性方程的一个特解y如何求？1xnexPxf其中xPn为n次多项式，为实常数，（1）若不是特征根，
14、则令xnexRy（2）若是特征方程单根，则令xnexxRy（3）若是特征方程的重根，则令xnexRxy22xexPxfxnsin或xexPxfxncos其中xPn为n次多项式，,皆为实常数（1）若i不是特征根，则令xxTxxReynnxsincos（2）若i是特征根，则令xxTxxRxeynnxsincos例题：一、齐次方程1.求dxdyxydxdyxy22的通解解：10)(22222xyxyxxyydxdydxdyxyxy令1,2uudxduxuuxy则0)1(duuxudx11Cxdxduuu，1|lnCuxu，xyuuCceyceexu,12.011dyyxedxeyxyx解：yxyxe
15、yxedydx11，令yuxuyx,.(将 y 看成自变量)dyduyudydx,所以uueuedyduyu1)1(uuuuueeuueeuedyduy11ydydueueuu1,ydyeueuduu)(,yyceuu1lnlnlnceuyu1,yxueyxceucy,cyexyx.二、一阶线形微分方程1.1)0(,0)(ydyxyydx解：可得0)1(1xyxdydx.这是以 y 为自变量的一阶线性方程解得)ln(ycyx.0)1(x,0c.所以得解yyxln.文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT
16、1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10
17、J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:C
18、K6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1
19、X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 Z
20、R3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R
21、5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N
22、4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X64 2.求微分方程4yxydxdy的通解解：变形得：341yxydydxyyxdydx即，是一阶线性方程3)
23、(,1)(yyQyyPCyyCdyeyexdyydyy413131三、伯努力方程63yxyxy解：356xyyxy，256xxyydxdy，令,5uy56uyy,25xxuu，255xuxu.解得)25(25xcxu，于是35525xcxy四、可降阶的高价微分方程1.求)1ln()1(xyyx的通解解：令pypy则,原方程化为)1ln()1(xppx1)1ln(11xxpxp属于一阶线性方程111111)1ln(Cdxexxepdxxdxx11)1ln()1ln(1111xCxCdxxx2111)1ln(CdxxCxy212)1ln()(CxxCx2.1)0(,2)0()(22yyyyy，解：
24、令dydppypy，则，得到ypdydpp22令up2,得到yudydu为关于 y 的一阶线性方程.1)0()0(0|22ypxu，解得yceyu1所以2)0(121)0(0|1ceceyxuy,0c.于是1yu,1ypdxydy1,112cxy,2211cxy2)0(y,得到121c,得解121xy五、二阶常系数齐次线形微分方程1.0 2 2)4()5(yyyyyy解：特征方程012223450)1)(1(22，ii5,43,21,1于是得解xxccxxccecyxc o s)(s i n)(543212.0610 5)4(yyyy，14)0(,6)0(,0)0(,1)0(yyyy解：特征方
25、程0610524,0)22)(3)(1(2文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J
26、10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK
27、6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X
28、6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR
29、3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5
30、F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4
31、 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10
32、S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X65 11,32,i14,3得通解为)s i nc o s(43321xcxceececyxxx由14)0(,6)0(,0)0(,1)0(yyyy得到211c,212c,13c,14c得特解)s in(c o s21213xxeeeyxxx六、二阶常系数非齐次线形微分方程1.求xeyyy232的通解解：先求齐次方程的通解，特征方程为0322，特征根为1,321。因此齐次方程通解为xxeCeCY231设非齐次方程的特解为1,由于y为特征根，因此设xxAey,代入原方程可得21A，故原方程的通解为xxxxeeCeCy212312.求方
33、程xyyycos222的通解解：特征方程为022，特征根为1,221，因此齐次方程的通解为xxeCeCY221设非齐次方程的特解为y,由于题目中ii2,2,0不是特征根，因此设xBxAy2sin2cos，代入原方程可得xxBABxABA2cos22sin)422(2cos)422(226BA，026AB解联立方程得101,103BA，因此xxy2sin1012cos103_故原方程的通解为xxeCeCyxx2sin1012cos1032213.xxxyycos22sin3 解：特征根为i，齐次方程的通解为：xcxcysincos21xyy ，xyccxccy1,02121xyy2sin3，xc
34、xcxcxcexyx2cos2sinsincos21210待入原式得出：0,121cc，所以xy2sinxyycos2，xxcxcxcxcexyx)sincos(sincos21211待入原式得出：1,021cc，所以xxysin故原方程的通解为xxxxxcxcysin2sinsincos21七、作变量代换后求方程的解1.求微分方程2322)1(1)(ydxdyxxy的通解解：令,tan,tanvxuy原方程化为uvdvuduvvu322secsecsecsec)tan(tan化简为1)sin(dvduvu再令方程化为则,1,dvdudvdzvuzzdvdzzsin1sin，,sin11)1(
35、sin,sin1sincvdzzzcdvdzzzcvdzzzz2sin1sin1，cvdzzzz2cossin1cvzzzsectan最后 Z 再返回 x,y,v 也返回 x,即可。文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8
36、ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3
37、R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10
38、N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U
39、10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文
40、档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q
41、9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F1
42、0O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X66 2.0)2(0)sin()1(yyxyx，解：设1dxdudxdyxuyuyxcxuuxdxuduudxduxlnlncotcsclnsin0sinxcyxyxxcuusincos1cotcsc，因为20,2cyx所以xyxyx2sincos13.21222sin22sin1xeyxyyx解：令yyuyu2sin,sin2则.得到212221xexuux,21221122xeuxxux为
43、一阶线性方程解得|)1|ln(2122xxceux.即|)1|ln(sin21222xxceyx.4.0)cos1(cossinlnyxyyxxy解：令uycos,则yyusin.原方程化为0)1(lnxuuxxuxuxxuulnln2,为贝奴利方程，xuxxuuln11ln12.令uz1,则2uuz.方程化为xzxxzln1ln1,为一阶线性方程.解得xcxzln)(.即xcxylncos1,xycxlncos)(.八、综合题1.设 f（x）xxsinxdttftx0)()(，其中 f（x）连续，求f（x）解：由表达式可知f（x）是可导的，两边对x 求导，则得xdttfxxxxf0sinco
44、s再对两边关于x 求导，得)(cos2sinxfxxxxf即xxxxfxfc o s2s i n属于常系数二阶非齐次线性方程.对应齐次方程通解xCxCysincos21，非 齐 次方 程 特解 设xDCxxxBAxxysincos_代入 方 程求 出 系数A,B,C,D 则得xxxxysin43cos412_，故 f（x）的一般表达式xCxCxxxxxfsincossin43cos41)(212由条件和导数表达式可知f（0）0，00f可确定出0,021CC因此xxxxxfsin43cos41)(22.已知xxexey21，xxexey2，xxxeexey23是某二阶线性非齐次常系数微分方程的三
45、个解，求此微分方程及其通解.解：由线性微分方程的解的结构定理可得，xeyy31，xxeeyy221，xeyyyy22131对应的齐次方程的解，由解xe与xe2的形式，可得齐次方程为02yyy.设该方程为)(2xfyyy，代入xxexey21，得xexxf21.所以，该方程为xexyyy212，其通解为xxxxexeeCeC2221.文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文
46、档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q
47、9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F1
48、0O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 H
49、T1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S1
50、0J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码:CK6U10S10J10N4 HT1H3R5F10O8 ZR3Q9N1M1X6文档编码: