電大常微分方程第二版答案第4章小抄參考

1、習 題 411求解下列微分方程1） 解 利用微分法得 當 時，得從而可得原方程的以p為參數的參數形式通解或消參數p，得通解 當 時，則消去p，得特解 2）； 解 利用微分法得 當時，得 從而可得原方程以p為參數的參數形式通解： 或消p得通解 當時，消去p得特解 3） 解 利用微分法，得 兩邊積分得由此得原方程以p為參數形式的通解： ，或消去p得通解1 用參數法求解下列微分方程1）解 將方程化為 令 由此可推出 從而得因此方程的通解為 ，消去參數t，得通解對于方程除了上述通解，還有，顯然和是方程的兩個解。2）解：令，又令 則積分得，由此得微分方程的通解為，3）解：令 則解得 又由此得微分方程的通

2、解為， 。習題421得用p判別式求下列方程的奇解：2）解：方程的p判別式為消去p，得經驗證可知是方程的解。令則有，和因此，由定理4.2可知， 是方程的奇解。2）解：方程的p判別式為，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判別式為 ，消去p，得，顯然是方程的解，令則有 和因此，由定理4.2知，是方程的奇解。2舉例說明，在定理4.2的條件中的兩個不等式是缺一不可的，解：考慮方程方程（1）的p判別式為 消去p，得 令，于是有 因此雖然有 和但是又雖然是方程的解，且容易求出方程（1）的通解為因此容易驗證卻不是奇解。因此由此例可看出。定理4.2中的條件是不可缺少的。又考慮方程

3、方程（2）的p判別式為 消去p，得。令于是有， 因此，雖然有和但，而經檢驗知是方程（2）的解，但不是奇解。因此由此例可看出定理4.2中的條件是不可缺少的。3研究下面的例子，說明定理4.2的條件是不可缺少的解：方程的p判別式為 消去p，得 檢驗知 不是解，故不是奇解，而雖然是解，但不是奇解。令， ， 所以雖有但是因此此例說明定理4.2的條件是不可缺少的。習題431試求克萊羅方程的通解及其包絡解：克萊羅方程 （1）其中。對方程（1）求導值由 即時 代入（1）得（1）的通解 （2）它的c判別式為 由此得 ， 令 故 所以 又 （由于）因此滿足定理4.5相應的非蛻化性條件。故是積分曲線族（2）的一支包

4、絡。課外補充1求下列給定曲線族的包絡。1）解：由相應的c判別式消去c得c判別曲線 它的兩支曲線的參數表示式為： ，： ，對，我們有因此滿足定理4.5的相應的非蛻化條件，同理可證，也滿足定理4.5的相應的非蛻化條件，故，是曲線族的兩支包絡線。2解：由相應的c判別式待添加的隱藏文字內容2消去c得c判別曲線 它的兩支曲線的參數表示式為 ， ，對，我們有 因此滿足定理4.5的相應的非蛻化條件，同理可證，也滿足定理4.5的相應的非蛻化條件，故，是曲線族的兩支包絡線。3. 證：就克萊羅方程來說，p判別曲線和方程通解的c判別曲線同樣是方程通解的包絡，從而為方程的奇解。證：已知克萊羅方程的形式為 （1）（1）

5、的通解為 （2）（2）的包絡由 確定，即為 （3）又知方程（1）還有解 由此得 ， （4）而（4）是方程（1）的p判別曲線，它和（3）有相同的形式，因而同樣是通解（2）的包絡，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 -

6、 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

7、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 v

8、illains 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 celeb

9、rations 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 pr

10、oceedings 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 th

11、e 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

12、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 eve

13、n 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 w

14、ill 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

15、 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

16、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