電大常微分方程第一次作業(yè)解析

1、常微分方程第一次作業(yè)解析1指出下列方程的階數(shù)，是否是線性方程：（1） （2） （3）解：（1）1階，是； （2）4階，是； （3）3階，不是。2用分離變量法求解下列方：（1） （2） （3） （4） 解：（1）通積分為：（2）當(dāng)時(shí)分離變量，兩端取積分得即 通積分為 另外，是常數(shù)解，注：在方程求解時(shí)，求出顯式通解或隱式通解（通積分）即可，常數(shù)解可以不求。（3）方法一：當(dāng)時(shí)，方程變?yōu)?積分得在通解中代入初值，有 . 所求特解為： 。 方法二：所求特解為,即 ，所求特解為： 。（4) 當(dāng)時(shí), 方程可變?yōu)橥ǚe分為 或 ,上式代入初值條件，得。 于是初值問題解為： 。3解下列齊次線性微分方程（1） （2

2、）（3） （4） 解：（1）顯然是方程的解。當(dāng)時(shí)，原方程可化為 。令， 則原方程可化為 ，即易于看出，是上面方程的解，從而，是原方程的解。當(dāng)時(shí)，分離變量得，. 兩端積分得：(c)。將換成，便得到原方程的解 ，(c)。故原方程的通解為（為任意常數(shù)）及 。（2） 顯然是方程的解。當(dāng)時(shí)，原方程可化為：。 令，則原方程可化為 ，即 易于看出，是上式的解，從而是原方程的解. 當(dāng)時(shí)，分離變量得 . 兩端積分得 (c). 將換成，便得到原方程的解 (c). 故原方程的通解為 . （3）顯然是方程的解. 當(dāng)時(shí)，原方程可化為令，則原方程可化為 ，即分離變量得，.兩端積分得 .將換成，便得到原方程的解 (c).

3、（4）將方程變形為: . 因?yàn)?，方程組 有解 ，令 . 代入原方程，得到新方程令，代入上式，又得到新方程 或當(dāng)時(shí)，有積分得原方程通積分為另外，由解得也是原方程解。4解下列一階線性微分方程：（1） （2）解：（1）先解齊次方程 。其通解為 . 用常數(shù)變易法，令非齊次方程通解為 . 代入原方程，化簡后可得 . 積分得到 . 代回后即得原方程通解為 . （2）先解齊次方程 . 其通解為 . 用常數(shù)變易法，令非齊次方程通解為 . 代入原方程，化簡后可得 . 積分得到 . 代回后即得原方程通解為 . 5解下列伯努利方程（1） （2） 解：（1）顯然是方程解. 當(dāng)時(shí)，兩端同除，得.令，代入有 ，它的解為

4、 。于是原方程的解為 ，及。（4）顯然是方程解。 當(dāng)時(shí)，兩端同除，得 . 令，代入有 它的解為 于是原方程的解 及 。6設(shè)函數(shù)，在上連續(xù)，且， （a, b為常數(shù)）求證：方程 的一切解在上有界 證：設(shè)y = y(x) 是方程任一解，且滿足y(x0)=y0，則由于，所以對任意0,存在x0，使得x時(shí)有令，則于是得到 又在x0，x1上y(x)有界設(shè)為m2，現(xiàn)取則 。7解下列全微分方程：（1） （2） 解：（1） 因?yàn)?，所以這方程是全微分方程，且及在整個(gè)平面都連續(xù)可微。不妨選取，故方程的通積分為 , 即 . （2）因?yàn)?，所以這方程是全微分方程，且及在整個(gè)平面都連續(xù)可微。不妨選取，故方程的通積分為 ,

5、 即 . 8求下列方程的積分因子和積分：（1） （2）解：因?yàn)?，與y無關(guān)，故原方程存在只含x的積分因子. 由公式（1.58）得積分因子即于是方程為全微分方程。取.于是方程的通積分為. 即 . （2）因?yàn)?，與y無關(guān)，故原方程存在只含x的積分因子。由公式（1.58）得積分因子即于是方程為全微分方程. 取，. 于是方程的通積分為. 即. 9求解下列一階隱式微分方程（1） （2）解：（1）將方程改寫為即 ，或解方程 ，得通積分為,又是常數(shù)解.（2） 顯然是方程的解. 當(dāng)時(shí)，方程可變?yōu)榱? 則上面的式子可變?yōu)?解出u得，. 即 . 對上式兩端積分得到方程的通解為. 10求解下列方程（1） （2） （

6、3） （4） 解：（1）令 ，則. 代入原式得.解出得.這是克萊洛方程，通解為.即.解之得（為任意常數(shù)）. （2）令 ，則, 原方程化為再令，則 即 從而 即 積分得通積分： . （3）化簡得 即 求積分得： ， 或 . （4）方程兩邊同乘以非零的數(shù), 化簡得.即 積分得再積分得11求曲線族的正交軌線，其中為參數(shù) 解：由 解該曲線所滿足的微分方程是：。于是正交軌線滿足的方程是： 即 積分得 為所求正交軌線： 。12人工繁殖細(xì)菌，其增長速度和當(dāng)時(shí)的細(xì)菌數(shù)成正比（1）如果過4小時(shí)的細(xì)菌數(shù)即為原細(xì)菌數(shù)的2倍，那么經(jīng)過12小時(shí)應(yīng)有多少？（2）如果在3小時(shí)的時(shí)候，有細(xì)菌數(shù)個(gè)，在5小時(shí)的時(shí)候有個(gè)，那么在開

7、始時(shí)有多少細(xì)菌？解：設(shè)時(shí)刻細(xì)菌數(shù)為，再設(shè)初值時(shí)刻時(shí)，細(xì)菌數(shù)為，則細(xì)菌繁殖滿足下列初值問題其中為正常數(shù)，初值問題為（1）由題意有即12小時(shí)后，細(xì)菌數(shù)為原來的8倍。（2）由題意有解得。13重為100kg的物體，在與水平面成30的斜面上由靜止?fàn)顟B(tài)下滑，如果不計(jì)磨擦，試求：（1）物體運(yùn)動的微分方程；（2）求5 s后物體下滑的距離，以及此時(shí)的速度和加速度解：取初始下滑點(diǎn)為原點(diǎn)，軸正向垂直向下，設(shè)時(shí)刻速度為 ，距離為，由題意滿足初值問題 解得 再由 ,，解得 。于是得到5秒后，。if we dont do that it will go on and go on. we have to stop it;

8、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 ivori

9、an 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 footba

10、ll 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

11、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

12、.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

13、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 a

14、s 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 tropos

15、phere) 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 tropo

16、pause),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, bu

17、t 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 t

18、han 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 p