常微分方程答案第三版习题3.4

1、习题 3.4(一)、解下列方程，并求奇解（如果存在的话）：1、解：令，则，两边对x求导，得 从得 时，；从得 ，为参数，为任意常数.经检验得，（）是方程奇解.2、解：令，则，两边对x求导，得 ，解之得 ，所以，且y=x+1也是方程的解，但不是奇解.3、解：这是克莱洛方程，因此它的通解为，从 中消去c,得到奇解.4、解：这是克莱洛方程，因此它的通解为 ，从 中消去c,得到奇解 .5、解：令，则，两边对x求导，得 ，解之得 ，所以 ，可知此方程没有奇解.6、解：原方程可化为，这是克莱罗方程，因此其通解为，从 中消去c，得奇解.7、解：令，则，两边对x求导，得 ，所以 ，可知此方程没有奇解.8、解：

2、可知此方程没有奇解.9、解：令，则，两边对x求导，得 解之得 ，所以 ，且 也是方程的解，但不是方程的奇解.10、解：这是克莱罗方程，因此方程的通解为，从中消去c,得方程的奇解.（二）求下列曲线族的包络.1、解:对c求导，得 x+2c=0, , 代入原方程得， 经检验得，是原方程的包络.2、解：对c求导，得 ，代入原方程得 ，即，经检验得是原方程的包络.3、解：对c求导，得 2(x-c)-2(y-c)=0, ,代入原方程得.经检验,得 是原方程的包络.4、解：对c求导，得 -2(x-c)=4, c=x+2,代入原方程得 ，,经检验，得是原方程的包络.(三) 求一曲线，使它上面的每一点的切线截割

3、坐标轴使两截距之和等于常数c.解：设所求曲线方程为y=y(x),以x、y表坐标系，则曲线上任一点（x,y(x)）的切线方程为，它与x轴、y轴的截距分别为，按条件有 ，化简得，这是克莱洛方程，它的通解为一族直线，它的包络是，消去c后得我们所求的曲线.(四) 试证：就克莱洛方程来说，p-判别曲线和方程通解的c-判别曲线同样是方程通解的包络，从而为方程的奇解.证：克莱洛方程 y=xp+f(p)的p-判别曲线就是用p-消去法，从 中消去p后而得的曲线； c-判别曲线就是用c-消去法，从通解及它对求导的所得的方程中消去c而得的曲线，显然它们的结果是一致的，是一单因式，因此p-判别曲线是通解的包络，也是方

4、程的通解.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 haverea

5、ched 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

6、 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 p

7、roblems 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,

8、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 cl