电大常微分方程第二版答案第三章小抄参考.doc

专业好文档习题31 1 判断下列方程在什么区域上保证初值解存在且唯一.1）； 2）； 3）.解 1）因为及在整个平面上连续，所以在整个平面上满足存在唯一性定理的条件，因此在整个平面上初值解存在且唯一.2）因为除轴外，在整个平面上连续，在在整个平面上有界，所以除轴外，在整个平面上初值解存在且唯一.3）设，则故在的任何有界闭区域上，及都连续，所以除轴外，在整个平面上初值解存在且唯一.2 求初值问题 R：.的解的存在区间.并求第二次近似解，给出在解的存在区间的误差估计.解 设，则，所以.显然，方程在R上满足解的存在唯一性定理，故过点的解的存在区间为：.设是方程的解，是第二次近似解，则，.在区间上，与的误差为 .取，故.3 讨论方程在怎样的区域中满足解的存在唯一性定理的条件.并求通过点的一切解.解 设，则.故在的任何有界闭区域上及都是连续的，因而方程在这种区域中满足解的存在唯一性定理的条件.显然，是过的一个解.又由解得.其中.所以通过点的一切解为及如图.4 试求初值问题，的毕卡序列，并由此取极限求解.解 按初值问题取零次近似为，一次近似为 ，二次近似为 ,三次近似为 ,四次近似为 ，五次近似为 ，一般地，利用数学归纳法可得次近似为 ，所以取极限得原方程的解为.5 设连续函数对是递减的，则初值问题，的右侧解是唯一的.证 设，是初值问题的两个解，令，则有.下面要证明的是当时，有.用反证法.假设当时，不恒等于0，即存在，使得，不妨设，由的连续性及，必有，使得，.又对于，有，则有，.由（）以及对是递减的，可以知道：上式左端大于零，而右端小于零.这一矛盾结果，说明假设不成立，即当时，有.从而证明方程的右侧解是唯一的.习题331 利用定理5证明：线性微分方程 （） 的每一个解的（最大）存在区间为，这里假设在区间上是连续的.证 在任何条形区域（其中）中连续，取，则有.故由定理5知道，方程的每一个解在区间中存在，由于是任意选取的，不难看出可被延拓到整个区间上.2 讨论下列微分方程解的存在区间： 1）； 2）； 3）.解 1）因在整个平面上连续可微，所以对任意初始点，方程满足初始条件的解存在唯一.这个方程的通解为.显然，均是该方程在上的解.现以，为界将整个平面分为三个区域来讨论.）在区域内任一点，方程满足的解存在唯一.由延伸定理知，它可以向左、右延伸，但不能与，两直线相交，因而解的存在区间为.又在内，则方程满足的解递减，当时，以为渐近线，当时，以为渐近线.）在区域中，对任意常数，由通解可推知，解的最大存在区间是，又由于，则对任意，方程满足的解递增.当时，以为渐近线，且每个最大解都有竖渐近线，每一条与轴垂直的直线皆为某解的竖渐近线.）在区域中，类似，对任意常数，解的最大存在区间是，又由于，则对任意，方程满足的解递增.当时，以为渐近线，且每个最大解都有竖渐近线.其积分曲线分布如图（ ）.2）因在整个平面上连续，且满足不等式，从而满足定理5的条件，故由定理5知，该方程的每一个解都以为最大存在区间.3）变量分离求得通解，故解的存在区间为.3设初值问题： ，的解的最大存在区间为，其中是平面上的任一点，则和中至少有一个成立.证明 因在整个平面上连续可微，所以对任意初始点，方程满足初始条件的解存在唯一.很显然，均是该方程在上的解.现以，为界将整个平面分为三个区域来进行讨论.）在区域内任一点，方程满足的解存在唯一.由延伸定理知，它可以向左、右延伸，但不能与，两直线相交，因而解的存在区间为.这里有，.）在区域中，由于，积分曲线单调上升.现设位于直线的下方，即，则利用的右行解的延伸定理，得出的解可以延伸到的边界.另一方面，直线的下方，积分曲线是单调上升的，并且它在向右延伸时不可能从直线穿越到上方.因此它必可向右延伸到区间.故至少成立.类似可证，对，至少有成立.4 设二元函数在全平面连续.求证：对任何，只要适当小，方程 的满足初值条件的解必可延拓到.证明 因为在全平面上连续，令，则在全平面上连续，且满足.对任何，选取，使之满足.设方程经过点的解为，在平面内延伸为方程的最大存在解时，它的最大存在区间为，由延伸定理可推知，或或为有限数且.下证后一种情形不可能出现.事实上，若不然，则必存在，使.不妨设.于是必存在，使，（）.此时必有，但，从而矛盾. 因此，即方程的解（）必可延拓到."If we don't 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 FA's 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 Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's 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 it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings against CSKA for the "racist behaviour of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed" by Toure's 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, he'll 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 it's 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 consciousness at mach speeds.Even if everything goes as planned, it won't. Baumgartner still will free fall at a speed that would cause you and me to pass out, and no parachute is guaranteed to work higher than 25,000 feet (7,620 meters).cause there8