2014年电大常微分方程第二版答案第6章6-2小抄参考.doc

专业好文档习题621 求出常系数齐次性微分方程组的通解，其中的矩阵a分别为1） 2） 3）4） 5）解：1） 特征方程 即 矩阵a有特征根， 对应于所有的特征向量满足即。取，则 那么对应的实值解为；对应的特征向量满足即，取，则，那么对应的实值解为 。于是该方程组的通解为2）特征方程为 即矩阵a有特征根 对应的特征向量应满足取，则 即么对应的特解为 由此得所对应的两个特解为（对2x2的方程组取一个特解的实部和虚部就可，因为虚根都是成对出现的。） 它们在上线性无关，故得方程组的通解：3） 即矩阵a有特征根 ，。对应于 ，特征向量应满足 又（只能进行行变换）因此与相应的特征向量可取为，对于二重特征根，可以算出因此，方程 有二个线性无关的解为 ，注意到，就可得到 从而可行基解矩阵 因此所求通解为 ，即 4）特征方程 即 矩阵a有特征根：，对应的特征向量应满足解之得 取 则故相应的解为 相应于 的特征向量应满足取 ，那么对应的复解为分别取实部，部可得方程组的两个实解,易知它们在上是线性无关的，于是方程组的通解为 5）特征方程为矩阵a的特征根为， 对应于，相应的特征向量应满足可以算出解之得， 则那么相应的解为 对应于三重特征根，可以算出因此，方程有三个线性无关解为， ， 注意到，可得由以上结果，可得方程组的一个基解矩阵因此所求方程组的通解为 或2求出常系数非齐次线性方程组，的通解，其中：3），； 4）， ；5）， 。3）解先求对应齐次方程组的通解特征方程 ，特征根为对于二重特征根，可以算出因此方程 有二个线性无关的解 由此可得齐次线性方程组的一个基解矩阵故非齐次方程组的通解为 容易求出故 于是非齐次方程组的通解为4）先求对应齐次方程组的通解特征方程为特征根为 ， ，对应于的特征向量为 对应于的特征向量为应满足 解之得 ，令，则其相应的复值解为：分别取实部和虚部，可得齐次方程组的两个线性无关的实解, 从而可得齐次方程组的一个基解矩阵 容易求得 这个矩阵的逆的算法：这里是只能通过行变换将矩阵先变成下三角，再变成对角阵即可。自己认真算，我都能算对，大家一定可以的，复习高等代数了。我仔细算了一下，要是将齐次方程的通解写出来，再用常数变易法求出特解方程组的阶数高的时候比求矩阵的逆还复杂，所以还是建议大家用求矩阵的逆的方法来算吧。故 则 所以非齐次线性方程组的通解为(5)先求对应齐次方程组的通解特征方程特征方程根为 。对于三重特重根 ，可以算出因此方程 有三个线性无关的解 ， 由此可得齐次线性方程的一个基解矩阵 从而容易求得 又 故 故非齐次线性方程组的通解为 由于特征向量取的不同，结果肯能也不一样。但是课本答案出现肯定是不正确的。3求出微分方程组 满足初值条件 的解，其中：（1）， ， ；（2）， ， ；（3）， ， 解 （1）齐次方程组的特征方程为 特征根 ： 对于二重特征根 ，可以算出 同此方程 有二个线性无关解， 由此可得齐次方程组的一个基解矩阵从而可求得 故 所以 故非齐次线性方程组的通解为由初始条件 解之得 ， 故初值问题的解为 （2）齐次方程组的特征方程为 特征根为 ， 对应 的特征向量应满足 取，则 故 从而可得齐次方程组的一个基解矩阵容易求得 而 又 故非齐次线性方程的通解为由初始条 有 解之得 故初值问题的解为 或 (3) 齐次方程组的特征方程为特征根 对应 的特征向量应满足 取 ,则 那么相应的解为 对应 的特征向量应满足, 取 ,则那么相应的解为 从而得齐次线性方程组的基解矩阵为 容易求得 由于 又 故非齐线性方程组通解为 由初值条件 ，得 解之得 因此值问题解为 或 4.证明:常系数齐次方程组 的任何解当 时都趋于零,当仅当它的系数矩阵a的所有特征根都具有负的实部.证 必要性:设特征根为 ,与之对应的方程组的解可表为 。1）当 即 为实数时，的每一分量或者为一常向量，或者为的多项式的向量函数。此时总有当 时, 或者是常向量。那么只有当 时, ,故必为负实数.2）当 时, 为复数, 则此时 其中 的向量多项式,当时, ,那么,若使当时,有成立,只有,于是,必为负实数。充分性：若系数矩阵a的所有特征根都具有负的实部，设特征根为，与之对应的解为 （1）当时，为负数，由解的结构知，是关于的一个多项式的向量函数，而已知 ，其中为任意自然数，故形如（1）的解当时，。（3）当时，是复数，由解的结构，此时（1）中的，其中是的多项式向量函数，又由于 故形如（1）的解，当时，。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 op