電大常微分方程第二版答案第6章62小抄參考

1、專業(yè)好文檔習(xí)題621 求出常系數(shù)齊次性微分方程組的通解，其中的矩陣a分別為1） 2） 3）4） 5）解：1） 特征方程 即 矩陣a有特征根， 對應(yīng)于所有的特征向量滿足即。取，則 那么對應(yīng)的實值解為；對應(yīng)的特征向量滿足即，取，則，那么對應(yīng)的實值解為 。于是該方程組的通解為2）特征方程為 即矩陣a有特征根 對應(yīng)的特征向量應(yīng)滿足取，則 即么對應(yīng)的特解為 由此得所對應(yīng)的兩個特解為（對2x2的方程組取一個特解的實部和虛部就可，因為虛根都是成對出現(xiàn)的。） 它們在上線性無關(guān)，故得方程組的通解：3） 即矩陣a有特征根 ，。對應(yīng)于 ，特征向量應(yīng)滿足 又（只能進(jìn)行行變換）因此與相應(yīng)的特征向量可取為，對于二重特征根

2、，可以算出因此，方程 有二個線性無關(guān)的解為 ，注意到，就可得到 從而可行基解矩陣 因此所求通解為 ，即 4）特征方程 即 矩陣a有特征根：，對應(yīng)的特征向量應(yīng)滿足解之得 取 則故相應(yīng)的解為 相應(yīng)于 的特征向量應(yīng)滿足取 ，那么對應(yīng)的復(fù)解為分別取實部，部可得方程組的兩個實解,易知它們在上是線性無關(guān)的，于是方程組的通解為 5）特征方程為矩陣a的特征根為， 對應(yīng)于，相應(yīng)的特征向量應(yīng)滿足可以算出解之得， 則那么相應(yīng)的解為 對應(yīng)于三重特征根，可以算出因此，方程有三個線性無關(guān)解為， ， 注意到，可得由以上結(jié)果，可得方程組的一個基解矩陣因此所求方程組的通解為 或2求出常系數(shù)非齊次線性方程組，的通解，其中：3），

3、； 4）， ；5）， 。3）解先求對應(yīng)齊次方程組的通解特征方程 ，特征根為對于二重特征根，可以算出因此方程 有二個線性無關(guān)的解 由此可得齊次線性方程組的一個基解矩陣故非齊次方程組的通解為 容易求出故 于是非齊次方程組的通解為4）先求對應(yīng)齊次方程組的通解特征方程為特征根為 ， ，對應(yīng)于的特征向量為 對應(yīng)于的特征向量為應(yīng)滿足 解之得 ，令，則其相應(yīng)的復(fù)值解為：分別取實部和虛部，可得齊次方程組的兩個線性無關(guān)的實解, 從而可得齊次方程組的一個基解矩陣 容易求得 這個矩陣的逆的算法：這里是只能通過行變換將矩陣先變成下三角，再變成對角陣即可。自己認(rèn)真算，我都能算對，大家一定可以的，復(fù)習(xí)高等代數(shù)了。我仔細(xì)算

4、了一下，要是將齊次方程的通解寫出來，再用常數(shù)變易法求出特解方程組的階數(shù)高的時候比求矩陣的逆還復(fù)雜，所以還是建議大家用求矩陣的逆的方法來算吧。故 則 所以非齊次線性方程組的通解為(5)先求對應(yīng)齊次方程組的通解特征方程特征方程根為 。對于三重特重根 ，可以算出因此方程 有三個線性無關(guān)的解 ， 由此可得齊次線性方程的一個基解矩陣 從而容易求得 又 故 故非齊次線性方程組的通解為 由于特征向量取的不同，結(jié)果肯能也不一樣。但是課本答案出現(xiàn)肯定是不正確的。3求出微分方程組 滿足初值條件 的解，其中：（1）， ， ；（2）， ， ；（3）， ， 解 （1）齊次方程組的特征方程為 特征根 ： 對于二重特征根

5、，可以算出 同此方程 有二個線性無關(guān)解， 由此可得齊次方程組的一個基解矩陣從而可求得 故 所以 故非齊次線性方程組的通解為由初始條件 解之得 ， 故初值問題的解為 （2）齊次方程組的特征方程為 特征根為 ， 對應(yīng) 的特征向量應(yīng)滿足 取，則 故 從而可得齊次方程組的一個基解矩陣容易求得 而 又 故非齊次線性方程的通解為由初始條 有 解之得 故初值問題的解為 或 (3) 齊次方程組的特征方程為特征根 對應(yīng) 的特征向量應(yīng)滿足 取 ,則 那么相應(yīng)的解為 對應(yīng) 的特征向量應(yīng)滿足, 取 ,則那么相應(yīng)的解為 從而得齊次線性方程組的基解矩陣為 容易求得 由于 又 故非齊線性方程組通解為 由初值條件 ，得 解之

6、得 因此值問題解為 或 4.證明:常系數(shù)齊次方程組 的任何解當(dāng) 時都趨于零,當(dāng)僅當(dāng)它的系數(shù)矩陣a的所有特征根都具有負(fù)的實部.證 必要性:設(shè)特征根為 ,與之對應(yīng)的方程組的解可表為 。1）當(dāng) 即 為實數(shù)時，的每一分量或者為一常向量，或者為的多項式的向量函數(shù)。此時總有當(dāng) 時, 或者是常向量。那么只有當(dāng) 時, ,故必為負(fù)實數(shù).2）當(dāng) 時, 為復(fù)數(shù), 則此時 其中 的向量多項式,當(dāng)時, ,那么,若使當(dāng)時,有成立,只有,于是,必為負(fù)實數(shù)。充分性：若系數(shù)矩陣a的所有特征根都具有負(fù)的實部，設(shè)特征根為，與之對應(yīng)的解為 （1）當(dāng)時，為負(fù)數(shù)，由解的結(jié)構(gòu)知，是關(guān)于的一個多項式的向量函數(shù)，而已知 ，其中為任意自然數(shù)，故

7、形如（1）的解當(dāng)時，。（3）當(dāng)時，是復(fù)數(shù)，由解的結(jié)構(gòu)，此時（1）中的，其中是的多項式向量函數(shù)，又由于 故形如（1）的解，當(dāng)時，。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 i

8、nternational 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

9、 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. bu

10、t 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

11、 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

12、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 rac

13、ist 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 humid

14、ity 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 o

15、f 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 re

16、sembling 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 shallo

17、w 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 comple

18、tely.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