2014年电大常微分方程第二版答案第4章小抄参考.doc

专业好文档习 题 411求解下列微分方程1） 解 利用微分法得 当 时，得从而可得原方程的以p为参数的参数形式通解或消参数p，得通解 当 时，则消去p，得特解 2）； 解 利用微分法得 当时，得 从而可得原方程以p为参数的参数形式通解： 或消p得通解 当时，消去p得特解 3） 解 利用微分法，得 两边积分得由此得原方程以p为参数形式的通解： ，或消去p得通解1 用参数法求解下列微分方程1）解 将方程化为 令 由此可推出 从而得因此方程的通解为 ，消去参数t，得通解对于方程除了上述通解，还有，显然和是方程的两个解。2）解：令，又令 则积分得，由此得微分方程的通解为，3）解：令 则解得 又由此得微分方程的通解为， 。习题421得用p判别式求下列方程的奇解：2）解：方程的p判别式为消去p，得经验证可知是方程的解。令则有，和因此，由定理4.2可知， 是方程的奇解。2）解：方程的p判别式为，消去p，得 ，而不是方程的解，故不是方程的奇解。3）解：方程的p判别式为 ，消去p，得，显然是方程的解，令则有 和因此，由定理4.2知，是方程的奇解。2举例说明，在定理4.2的条件中的两个不等式是缺一不可的，解：考虑方程方程（1）的p判别式为 消去p，得 令，于是有 因此虽然有 和但是又虽然是方程的解，且容易求出方程（1）的通解为因此容易验证却不是奇解。因此由此例可看出。定理4.2中的条件是不可缺少的。又考虑方程 方程（2）的p判别式为 消去p，得。令于是有， 因此，虽然有和但，而经检验知是方程（2）的解，但不是奇解。因此由此例可看出定理4.2中的条件是不可缺少的。3研究下面的例子，说明定理4.2的条件是不可缺少的解：方程的p判别式为 消去p，得 检验知 不是解，故不是奇解，而虽然是解，但不是奇解。令， ， 所以虽有但是因此此例说明定理4.2的条件是不可缺少的。习题431试求克莱罗方程的通解及其包络解：克莱罗方程 （1）其中。对方程（1）求导值由 即时 代入（1）得（1）的通解 （2）它的c判别式为 由此得 ， 令 故 所以 又 （由于）因此满足定理4.5相应的非蜕化性条件。故是积分曲线族（2）的一支包络。课外补充1求下列给定曲线族的包络。1）解：由相应的c判别式消去c得c判别曲线 它的两支曲线的参数表示式为： ，： ，对，我们有因此满足定理4.5的相应的非蜕化条件，同理可证，也满足定理4.5的相应的非蜕化条件，故，是曲线族的两支包络线。2解：由相应的c判别式消去c得c判别曲线 它的两支曲线的参数表示式为 ， ，对，我们有 因此满足定理4.5的相应的非蜕化条件，同理可证，也满足定理4.5的相应的非蜕化条件，故，是曲线族的两支包络线。3. 证：就克莱罗方程来说，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：已知克莱罗方程的形式为 （1）（1）的通解为 （2）（2）的包络由 确定，即为 （3）又知方程（1）还有解 由此得 ， （4）而（4）是方程（1）的p判别曲线，它和（3）有相同的形式，因而同样是通解（2）的包络，消去p得方程（1）的奇解。if we dont do that it will go on and go on. we have to stop it; we need the courage to do it.his comments came hours after fifa vice-president jeffrey webb - also in london for the fas celebrations - said he wanted to meet ivory coast international toure to discuss his complaint.cska general director roman babaev says the matter has been exaggerated by the ivorian and the british media.blatter, 77, said: it has been decided by the fifa congress that it is a nonsense for racism to be dealt with with fines. you can always find money from somebody to pay them.it is a nonsense to have matches played without spectators because it is against the spirit of football and against the visiting team. it is all nonsense.we can do something better to fight racism and discrimination.this is one of the villains we have today in our game. but it is only with harsh sanctions that racism and discrimination can be washed out of football.the (lack of) air up there watch mcayman islands-based webb, the head of fifas anti-racism taskforce, is in london for the football associations 150th anniversary celebrations and will attend citys premier league match at chelsea on sunday.i am going to be at the match tomorrow and i have asked to meet yaya toure, he told bbc sport.for me its about how he felt and i would like to speak to him first to find out what his experience was.uefa hasopened disciplinary proceedings against cskafor the racist behaviour of their fans duringcitys 2-1 win.michel platini, president of european footballs governing body, has also ordered an immediate investigation into the referees actions.cska said they were surprised and disappointed by toures complaint. in a statement the russian side added: we found no racist insults from fans of cska.baumgartner the disappointing news: mission aborted.the supersonic descent could happen as early as sunda.the weather plays an important role in this mission. starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. the balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. it will climb higher than the tip of mount everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. as he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.the balloon will slowly drift to the edge of space at 120,000 feet ( then, i would assume, he will slowly step out onto something resembling an olympic diving platform.below, the earth becomes the concrete bottom of a swimming pool that he wants to land on, but not too hard. still, hell be traveling fast, so despite the distance, it will not be like diving into the deep end of a pool. it will be like he is diving into the shallow end.skydiver preps for the big jumpwhen he jumps, he is expected to reach the speed of sound - 690 mph (1,110 kph) - in less than 40 seconds. like hitting the top of the water, he will begin to slow as he approaches the more dense air closer to earth. but this will not be enough to stop him completely.if he goes too fast or spins out of control, he has a stabilization parachute that can be deployed to slow him down. his team hopes its not needed. instead, he plans to deploy his 270-square-foot (25-square-meter) main chute at an altitude of around 5,000 feet (1,524 meters).in order to deploy this chute successfully, he will have to slow to 172 mph (277 kph). he will have a reserve parachute that will open automatically if he loses