常微分方程课后答案(第三版).doc

习题1.21=2xy,并满足初始条件：x=0,y=1的特解。解：=2xdx 两边积分有：ln|y|=x+cy=e+e=cex另外y=0也是原方程的解，c=0时，y=0原方程的通解为y= cex,x=0 y=1时 c=1特解为y= e.2. ydx+(x+1)dy=0 并求满足初始条件：x=0,y=1的特解。 解：ydx=-(x+1)dy dy=-dx两边积分: -=-ln|x+1|+ln|c| y=另外y=0,x=-1也是原方程的解 x=0,y=1时 c=e特解：y=3= 解：原方程为：=dy=dx 两边积分：x(1+x)(1+y)=cx4. (1+x)ydx+(1-y)xdy=0 解：原方程为： dy=-dx两边积分：ln|xy|+x-y=c另外 x=0,y=0也是原方程的解。5（y+x）dy+(x-y)dx=0 解：原方程为： =-令=u 则=u+x 代入有：-du=dxln(u+1)x=c-2arctgu即 ln(y+x)=c-2arctg.6. x-y+=0 解：原方程为： =+-则令=u =u+ x du=sgnx dxarcsin=sgnx ln|x|+c7. tgydx-ctgxdy=0 解:原方程为：=两边积分：ln|siny|=-ln|cosx|-ln|c|siny= 另外y=0也是原方程的解，而c=0时，y=0.所以原方程的通解为sinycosx=c.8 +=0 解：原方程为：=e2 e-3e=c.9.x(lnx-lny)dy-ydx=0 解：原方程为：=ln令=u ,则=u+ xu+ x=ulnuln(lnu-1)=-ln|cx|1+ln=cy.10. =e 解：原方程为：=eee=ce11 =(x+y) 解：令x+y=u,则=-1-1=udu=dxarctgu=x+carctg(x+y)=x+c12. =解：令x+y=u,则=-1 -1= u-arctgu=x+c y-arctg(x+y)=c.13. =解: 原方程为：（x-2y+1）dy=(2x-y+1)dx xdy+ydx-(2y-1)dy-(2x+1)dx=0 dxy-d(y-y)-dx+x=c xy-y+y-x-x=c14: =解：原方程为：（x-y-2）dy=(x-y+5)dx xdy+ydx-(y+2)dy-(x+5)dx=0 dxy-d(y+2y)-d(x+5x)=0 y+4y+x+10x-2xy=c.15: =(x+1) +(4y+1) +8xy 解：原方程为：=（x+4y）+3令x+4y=u 则=-=u+3=4 u+13u=tg(6x+c)-1tg(6x+c)=(x+4y+1).16:证明方程=f(xy),经变换xy=u可化为变量分离方程，并由此求下列方程：1） y(1+xy)dx=xdy2） = 证明： 令xy=u,则x+y= 则=-，有： =f(u)+1 du=dx 所以原方程可化为变量分离方程。1） 令xy=u 则=- (1)原方程可化为：=1+（xy） (2)将1代入2式有：-=(1+u)u=+cx17.求一曲线，使它的切线坐标轴间的部分初切点分成相等的部分。解：设（x +y ）为所求曲线上任意一点，则切线方程为：y=y(x- x )+ y 则与x轴，y轴交点分别为： x= x - y= y - x y 则 x=2 x = x - 所以 xy=c18.求曲线上任意一点切线与该点的向径夹角为0的曲线方程，其中 = 。解：由题意得：y= dy= dx ln|y|=ln|xc| y=cx. = 则y=tgx 所以 c=1 y=x.19.证明曲线上的切线的斜率与切点的横坐标成正比的曲线是抛物线。 证明：设(x,y)为所求曲线上的任意一点，则y=kx 则：y=kx +c 即为所求。 acknowledgements my deepest gratitude goes first and foremost to professor aaa , my supervisor, for her constant encouragement and guidance. she has walked me through all the stages of the writing of this thesis. without her consistent and illuminating instruction, this thesis could not havereached its present form. second, i would like to express my heartfelt gratitude to professor aaa, who led me into the world of translation. i am also greatly indebted to the professors and teachers at the department of english: professor dddd, professor ssss, who have instructed and helped me a lot in the past two years. last my thanks would go to my beloved family for their loving considerations and great confidence in me all through these years. i also owe my sincere gratitude to my friends and my fellow classmates who gave me their help and time in listening to me and helping me work out my problems during the difficult course of the thesis. my deepest gratitude goes first and foremost to professor aaa , my supervisor, for her constant encouragement and guidance. she has walked me through all the stages of the writing of this thesis. without her consistent and illuminating instruction, this thesis could not havereached its present form. second, i would like to express my heartfelt gratitude to professor aaa, who led me into the world of translation. i am also greatly indebted to the professors and teachers at the department of english: professor dddd, professor ssss, who have instructed and helped me a lot in the past two years. last my thanks would go to my beloved family for their loving considerations and great confidence in me all through these years. i also owe my sincere gratitu