常微分方程第三版答案2.2

1、习题2.2求下列方程的解1=解： y=e (e)=e-e()+c=c e- ()是原方程的解。2+3x=e解：原方程可化为：=-3x+e所以：x=e (e e) =e (e+c) =c e+e 是原方程的解。3=-s+解：s=e(e )=e()= e()= 是原方程的解。4 ， n为常数.解：原方程可化为： 是原方程的解.5+=解：原方程可化为：=- ()= 是原方程的解.6 解： =+令 则 =u因此：= （*） 将带入 （*）中 得：是原方程的解.313这是n=-1时的伯努利方程。两边同除以，令 p(x)= q(x)=-1由一阶线性方程的求解公式 =14 两边同乘以 令 这是n=2时的伯努

2、利方程。两边同除以 令 p（x）= q(x)=由一阶线性方程的求解公式 = =15 这是n=3时的伯努利方程。两边同除以 令 = p(y)=-2y q(y)= 由一阶线性方程的求解公式 =16 y=+p(x)=1 q(x)= 由一阶线性方程的求解公式 = =c=1y=17 设函数(t)于t上连续，(0)存在且满足关系式(t+s)=(t)(s)试求此函数。令t=s=0 得(0+0)=(0)(0) 即(0)= 故或（1） 当时 即 ,) (2) 当时 = = =于是 变量分离得 积分 由于，即t=0时 1=c=1故 20.试证： （1）一阶非齐线性方程（2 .28）的任两解之差必为相应的齐线性方程

3、（2.3）之解； （2）若是（2.3）的非零解，而是（2.28）的解，则方程（2.28）的通解可表为，其中为任意常数.（3）方程（2.3）任一解的常数倍或任两解之和（或差）仍是方程（2.3）的解.证明： （2.28） （2.3）（1） 设，是（2.28）的任意两个解则 （1） （2）（1）-（2）得 即是满足方程（2.3）所以，命题成立。（2） 由题意得： （3） （4）1）先证是（2.28）的一个解。于是 得故是（2.28）的一个解。2）现证方程（4）的任一解都可写成的形式设是(2.28)的一个解则 （4）于是 （4）-（4）得从而 即 所以，命题成立。（3） 设，是（2.3）的任意两个解则

4、 （5） （6）于是（5）得 即 其中为任意常数也就是满足方程（2.3）（5）（6）得 即 也就是满足方程（2.3）所以命题成立。21.试建立分别具有下列性质的曲线所满足的微分方程并求解。（5） 曲线上任一点的切线的纵截距等于切点横坐标的平方；（6） 曲线上任一点的切线的纵截距是切点横坐标和纵坐标的等差中项；解：设为曲线上的任一点，则过点曲线的切线方程为从而此切线与两坐标轴的交点坐标为即 横截距为 ， 纵截距为 。由题意得：（5） 方程变形为 于是 所以，方程的通解为。（6）方程变形为 于是 所以，方程的通解为。22求解下列方程。（1）解： = = = （2） p(x)= q(x)=由一阶线性

5、方程的求解公式 = = =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

6、 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

7、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

8、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 instr

9、uction, 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