微分方程与差分方程第八章微分方程与差分方程一、作业题1．«Skip Record If...»«Skip Record If...»«Skip Record If...»，«Skip Record If...»为任意常数(2)«Skip Record If...»设«Skip Record If...»，«Skip Record If...»，«Skip Record If...» (代入上式) «Skip Record If...»«Skip Record If...»，«Skip Record If...»«Skip Record If...»，«Skip Record If...»(3)«Skip Record If...»«Skip Record If...»«Skip Record If...»(4)«Skip Record If...»«Skip Record If...»«Skip Record If...»满足«Skip Record If...»的特解为«Skip Record If...»(5)设«Skip Record If...»代入（1）式中，«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»满足初始条件的特解为«Skip Record If...»(6)特征方程为«Skip Record If...»，解得«Skip Record If...»仅供学习与交流，如有侵权请联系网站删除谢谢70故方程的两个特解为«Skip Record If...»得到通解«Skip Record If...»因此方程满足初始条件«Skip Record If...»，故«Skip Record If...»所以满足初始条件的特解为«Skip Record If...»2.依题意，«Skip Record If...» («Skip Record If...»为常数)，且初始条件为«Skip Record If...»故解微分方程«Skip Record If...»，两边同时积分«Skip Record If...»得«Skip Record If...»，«Skip Record If...»，代入初始条件，«Skip Record If...»，故«Skip Record If...»、二、练习题1.填空（1）«Skip Record If...»（«Skip Record If...»为任意常数）（2）«Skip Record If...»（3）2（4）«Skip Record If...»（5）«Skip Record If...»2.选择（1）D（2）C（3）A（4）C（5）C3.求下列微分方程的解：仅供学习与交流，如有侵权请联系网站删除谢谢70（1）«Skip Record If...»«Skip Record If...»，故«Skip Record If...»即满足初始条件的特解为«Skip Record If...»（2）设«Skip Record If...»«Skip Record If...»«Skip Record If...»又由初始条件«Skip Record If...» «Skip Record If...»得«Skip Record If...»，特解为«Skip Record If...»（3）解：该方程为一阶线性非齐次方程，其中«Skip Record If...»，«Skip Record If...»方程的通解为：«Skip Record If...»«Skip Record If...»«Skip Record If...»（4）解：该方程可化为齐次方程 «Skip Record If...»，令«Skip Record If...»，则«Skip Record If...»，«Skip Record If...»，代入上式得«Skip Record If...» «Skip Record If...»«Skip Record If...» «Skip Record If...»«Skip Record If...»即为方程的通解（其中«Skip Record If...»）（5）解：特征方程为«Skip Record If...»，即«Skip Record If...»特征根为«Skip Record If...»，«Skip Record If...»，故方程的通解为 «Skip Record If...»（6）解：特征方程为«Skip Record If...»，即«Skip Record If...»特征根为«Skip Record If...»，故方程的通解为 «Skip Record If...»仅供学习与交流，如有侵权请联系网站删除谢谢70(7)解：特征方程为«Skip Record If...»，特征根为«Skip Record If...»，故方程的通解为 «Skip Record If...»代入初始条件«Skip Record If...»，得«Skip Record If...»，«Skip Record If...»所以方程的特解为«Skip Record If...»4．解：由题意得 «Skip Record If...»，则«Skip Record If...»方程为一阶线性非齐次方程，其中«Skip Record If...»，«Skip Record If...»方程的通解为：«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»又因为«Skip Record If...»，解得«Skip Record If...»，从而原方程的解为«Skip Record If...»5．解：由题意有«Skip Record If...»方程为可分离变量微分方程，分离变量得«Skip Record If...»两边积分求得通解为 «Skip Record If...»又因为«Skip Record If...»，解得«Skip Record If...»，故«Skip Record If...»为所求6．解：方程可化为 «Skip Record If...»方程两边同时对«Skip Record If...»求导得«Skip Record If...»«Skip Record If...»«Skip Record If...»（*）令方程 «Skip Record If...»中«Skip Record If...»，得«Skip Record If...»，代入（*）式有«Skip Record If...»，故«Skip Record If...»7．解：由题意有«Skip Record If...»方程为可分离变量微分方程，仅供学习与交流，如有侵权请联系网站删除谢谢70分离变量得 «Skip Record If...»，两边积分求得 «Skip Record If...»整理得«Skip Record If...»，代入«Skip Record If...»得«Skip Record If...»；代入«Skip Record If...»得«Skip Record If...»，所以方程的解为«Skip Record If...»故当«Skip Record If...»时，«Skip Record If...»（尾）三、提高题1．解：由题意得 «Skip Record If...»，两边对«Skip Record If...»求导得 «Skip Record If...»整理得 «Skip Record If...»2．解：方程«Skip Record If...»两边同时对«Skip Record If...»求导得«Skip Record If...»整理得«Skip Record If...»3．求下列微分方程的通解（1）解：该方程可化为齐次方程 «Skip Record If...»，令«Skip Record If...»，则«Skip Record If...»，«Skip Record If...»，代入上式得«Skip Record If...»分离变量得 «Skip Record If...»«Skip Record If...»即 «Skip Record If...»（令«Skip Record If...»）再将 «Skip Record If...»代入上式得方程的通解 «Skip Record If...»(2) 解：方程可化为一阶线性非齐次方程 «Skip Record If...»，其中«Skip RecordIf...»，«Skip Record If...»方程的通解为：«Skip Record If...»«Skip Record If...»仅供学习与交流，如有侵权请联系网站删除谢谢70«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»(3) 解：方程可化为一阶线性非齐次方程 «Skip Record If...»，其中«Skip Record If...»，«Skip Record If...»方程的通解为：«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»代入初始条件«Skip Record If...»时«Skip Record If...»，解得 «Skip Record If...»，故«Skip Record If...»（4）解：该方程为齐次方程，设«Skip Record If...»，则«Skip Record If...»，«Skip Record If...»，代入上式得 «Skip Record If...»分离变量得 «Skip Record If...» «Skip Record If...»再将 «Skip Record If...»代入上式得方程的通解 «Skip Record If...»，又因为 «Skip Record If...»，解得«Skip Record If...»，从而«Skip Record If...»，即«Skip Record If...»（5）解：方程可化为一阶线性非齐次方程 «Skip Record If...»，其中«Skip Record If...»，«Skip Record If...»方程的通解为：«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»仅供学习与交流，如有侵权请联系网站删除谢谢704．解：设质点运动的速度为«Skip Record If...»，由题意«Skip Record If...»方程整理为一阶线性非齐次方程为 «Skip Record If...»其中«Skip Record If...»，«Skip Record If...»方程的通解为：«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»5．解：由题意«Skip Record If...»，（«Skip Record If...»），解得«Skip Record If...»由«Skip Record If...»，得«Skip Record If...»，又因为«Skip Record If...»，解得«Skip Record If...»综上，«Skip Record If...»6．解：销售函数为«Skip Record If...»，«Skip Record If...»为最大销售量，则由题意有«Skip Record If...»，该方程为可分离变量微分方程，分离变量得 «Skip Record If...»，两边积分得 «Skip Record If...»«Skip Record If...»仅供学习与交流，如有侵权请联系网站删除谢谢70。