電大常微分方程第二版答案第三章小抄參考

1、專業(yè)好文檔習(xí)題31 1 判斷下列方程在什么區(qū)域上保證初值解存在且唯一.1）； 2）； 3）.解 1）因?yàn)榧霸谡麄€(gè)平面上連續(xù)，所以在整個(gè)平面上滿足存在唯一性定理的條件，因此在整個(gè)平面上初值解存在且唯一.2）因?yàn)槌S外，在整個(gè)平面上連續(xù)，在在整個(gè)平面上有界，所以除軸外，在整個(gè)平面上初值解存在且唯一.3）設(shè)，則故在的任何有界閉區(qū)域上，及都連續(xù)，所以除軸外，在整個(gè)平面上初值解存在且唯一.2 求初值問題 r：.的解的存在區(qū)間.并求第二次近似解，給出在解的存在區(qū)間的誤差估計(jì).解 設(shè)，則，所以.顯然，方程在r上滿足解的存在唯一性定理，故過點(diǎn)的解的存在區(qū)間為：.設(shè)是方程的解，是第二次近似解，則，.在區(qū)間上，與

2、的誤差為 .取，故.3 討論方程在怎樣的區(qū)域中滿足解的存在唯一性定理的條件.并求通過點(diǎn)的一切解.解 設(shè)，則.故在的任何有界閉區(qū)域上及都是連續(xù)的，因而方程在這種區(qū)域中滿足解的存在唯一性定理的條件.顯然，是過的一個(gè)解.又由解得.其中.所以通過點(diǎn)的一切解為及如圖.4 試求初值問題，的畢卡序列，并由此取極限求解.解 按初值問題取零次近似為，一次近似為 ，二次近似為 ,三次近似為 ,四次近似為 ，五次近似為 ，一般地，利用數(shù)學(xué)歸納法可得次近似為 ，所以取極限得原方程的解為.5 設(shè)連續(xù)函數(shù)對(duì)是遞減的，則初值問題，的右側(cè)解是唯一的.證 設(shè)，是初值問題的兩個(gè)解，令，則有.下面要證明的是當(dāng)時(shí)，有.用反證法.假設(shè)

3、當(dāng)時(shí)，不恒等于0，即存在，使得，不妨設(shè)，由的連續(xù)性及，必有，使得，.又對(duì)于，有，則有，.由（）以及對(duì)是遞減的，可以知道：上式左端大于零，而右端小于零.這一矛盾結(jié)果，說明假設(shè)不成立，即當(dāng)時(shí)，有.從而證明方程的右側(cè)解是唯一的.習(xí)題331 利用定理5證明：線性微分方程 （） 的每一個(gè)解的（最大）存在區(qū)間為，這里假設(shè)在區(qū)間上是連續(xù)的.證 在任何條形區(qū)域（其中）中連續(xù)，取，則有.故由定理5知道，方程的每一個(gè)解在區(qū)間中存在，由于是任意選取的，不難看出可被延拓到整個(gè)區(qū)間上.2 討論下列微分方程解的存在區(qū)間： 1）； 2）； 3）.解 1）因在整個(gè)平面上連續(xù)可微，所以對(duì)任意初始點(diǎn)，方程滿足初始條件的解存在唯一

4、.這個(gè)方程的通解為.顯然，均是該方程在上的解.現(xiàn)以，為界將整個(gè)平面分為三個(gè)區(qū)域來討論.）在區(qū)域內(nèi)任一點(diǎn)，方程滿足的解存在唯一.由延伸定理知，它可以向左、右延伸，但不能與，兩直線相交，因而解的存在區(qū)間為.又在內(nèi)，則方程滿足的解遞減，當(dāng)時(shí)，以為漸近線，當(dāng)時(shí)，以為漸近線.）在區(qū)域中，對(duì)任意常數(shù)，由通解可推知，解的最大存在區(qū)間是，又由于，則對(duì)任意，方程滿足的解遞增.當(dāng)時(shí)，以為漸近線，且每個(gè)最大解都有豎漸近線，每一條與軸垂直的直線皆為某解的豎漸近線.）在區(qū)域中，類似，對(duì)任意常數(shù)，解的最大存在區(qū)間是，又由于，則對(duì)任意，方程滿足的解遞增.當(dāng)時(shí)，以為漸近線，且每個(gè)最大解都有豎漸近線.其積分曲線分布如圖（ ）.

5、2）因在整個(gè)平面上連續(xù)，且滿足不等式，從而滿足定理5的條件，故由定理5知，該方程的每一個(gè)解都以為最大存在區(qū)間.3）變量分離求得通解，故解的存在區(qū)間為.3設(shè)初值問題： ，的解的最大存在區(qū)間為，其中是平面上的任一點(diǎn)，則和中至少有一個(gè)成立.證明 因在整個(gè)平面上連續(xù)可微，所以對(duì)任意初始點(diǎn)，方程滿足初始條件的解存在唯一.很顯然，均是該方程在上的解.現(xiàn)以，為界將整個(gè)平面分為三個(gè)區(qū)域來進(jìn)行討論.）在區(qū)域內(nèi)任一點(diǎn)，方程滿足的解存在唯一.由延伸定理知，它可以向左、右延伸，但不能與，兩直線相交，因而解的存在區(qū)間為.這里有，.）在區(qū)域中，由于，積分曲線單調(diào)上升.現(xiàn)設(shè)位于直線的下方，即，則利用的右行解的延伸定理，得出

6、的解可以延伸到的邊界.另一方面，直線的下方，積分曲線是單調(diào)上升的，并且它在向右延伸時(shí)不可能從直線穿越到上方.因此它必可向右延伸到區(qū)間.故至少成立.類似可證，對(duì)，至少有成立.4 設(shè)二元函數(shù)在全平面連續(xù).求證：對(duì)任何，只要適當(dāng)小，方程 的滿足初值條件的解必可延拓到.證明 因?yàn)樵谌矫嫔线B續(xù)，令，則在全平面上連續(xù)，且滿足.對(duì)任何，選取，使之滿足.設(shè)方程經(jīng)過點(diǎn)的解為，在平面內(nèi)延伸為方程的最大存在解時(shí)，它的最大存在區(qū)間為，由延伸定理可推知，或或?yàn)橛邢迶?shù)且.下證后一種情形不可能出現(xiàn).事實(shí)上，若不然，則必存在，使.不妨設(shè).于是必存在，使，（）.此時(shí)必有，但，從而矛盾. 因此，即方程的解（）必可延拓到.if

7、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 r

8、oman 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

9、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 ca

10、n 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 to

11、morrow 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, preside

12、nt 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: m

13、ission 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, wi

14、ll 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 str

15、atosphere. 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 co

16、ncrete 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

17、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 stabiliz

18、ation 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