专业好文档浙江省2010年4月高等教育自学考试常微分方程试题课程代码：10002本试卷分A、B卷，使用1983年版本教材的考生请做A卷，使用2006年版本教材的考生请做B卷；若A、B两卷都做的，以B卷记分。A卷一、填空题(本大题共11小题，每空3分，共36分)请在每小题的空格中填上正确答案。错填、不填均无分。1.方程有只含x的积分因子的充要条件是_,有只含y的积分因子的充要条件是_.2. 方程通过点(6,-1)的解的存在区间为_.3. 如果存在常数L0，使得不等式_对于所有都成立，则函数称为在R上关于y满足利普希兹条件. 4. 方程y-x2y+3y=0的解空间构成_ .5. 若xi(t)(i=1,2,n)为齐线性方程的一个基本解组，(t)为对应的非齐线性方程的一个特解，则非齐线性方程的所有解可表为 _.6. 方程的所有常数解是_.7.方程的基本解组是_.8.n阶非齐线性方程最多存在_个线性无关的解.9.A是一个nn常数矩阵，矩阵(exp At)=_.10.若矩阵A具有n个线性无关的特征向量v1,v2,vn，它们对应的特征值分别为1,2，n,那么矩阵(t)=_是常系数线性方程组x=Ax的一个基解矩阵.11.方程组满足_的点(x*,y*)称为此方程组的奇点.二、计算题（本大题共7小题，每小题8分，共56分）1.解方程：.2.解方程：(2x+2y-1)dx+(x+y-2)dy=0.3.求解常系数线性方程：x-2x+3x=e-tcost.4.求方程的通解.5.试求方程组x=Ax的一个基解矩阵，并计算exp At，其中A=6.求=-x-y+1, =x-y-5的奇点,并判断奇点的类型及稳定性.7.判断方程组的零解的稳定性.三、证明题（本大题8分）假设m不是矩阵A的特征值，试证非齐线性方程组x=Ax+cemt有一解形如(t)=pemt其中c,p是常数向量.B卷一、填空题(本大题共12小题，每小题3分，共36分)请在每小题的空格中填上正确答案。错填、不填均无分。1.方程+cosy+2x=0是一阶_方程.2.方程=2x的通解为_.3.方程=g()称为_微分方程,其中g(u)是u的连续函数.4.若函数f(x,y)以及都在区域G内连续,则方程=f(x,y)的解y=(x,x0,y0)作为x,x0,y0的函数在它的存在范围内是_.5.初值问题的第一次近似解为_.6.n阶齐次线性微分方程的n个解构成的朗斯基行列式或者_,或者在方程的系数为连续的区间内处处不等于零.7.n阶齐次线性微分方程的一组n个线性无关解称为方程的一个_.8.方程x=的解为_.9.如果T是非奇异矩阵,则exp(T-1AT)=_.10.n次多项式p()=det(E-A)称为A的_,其中E,A是nn矩阵.11.函数V(x,y)=x2-2xy2的定号性是_.12.方程组的唯一奇点(0,0)的类型是_.二、计算题(本大题共8小题，每小题6分，共48分)13.求解方程14.求解方程(x+2y)dx+xdy=0.15.求解方程=-.16.求解方程x2+(y)2=1.17.求方程x(4)-2x+x=t2-3的通解.18.求方程y-y=的通解.19.求方程组的一个基解矩阵.20.用李雅普诺夫函数讨论方程组零解的稳定性,其中为参数.三、证明题(本大题共2小题，每小题8分，共16分)21.设K为非负常数,f(t)和g(t)为在区间t上的连续非负函数,且满足不等式f(t)K+(f(s)g(s)ds,t,则有f(t)K expg(s)ds),t.22.证明:对于微分方程x+p(t)x+q(t)x=0,其中p(t),q(t)为连续函数,若存在常数m使得m2+mp(t)+q(t)0,则方程有解x=emt.Winger Tuivasa-Sheck, who scored two tries in the Kiwis 20-18 semi-final win over England, has been passed fit after a lower-leg injury, while Slater has been named at full-back but is still recovering from a knee injury aggravated against USA.Both sides boast 100% records heading into the encounter but Australia have not conceded a try since Josh Charnleys effort in their first pool match against England on the opening day.Aussie winger Jarryd Hayne is the competitions top try scorer with nine, closely followed by Tuivasa-Sheck with eight.But it is recently named Rugby League International Federation player of the year Sonny Bill Williams who has attracted the most interest in the tournament so far.The Kiwi - with a tournament high 17 offloads - has the chance of becoming the first player to win the World Cup in both rugby league and rugby union after triumphing with the All Blacks in 2011.Id give every award back in a heartbeat just to get across the line this weekend, said Williams.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.They blew it in 2008 when they got caught cold in the final and they will not make the same mistake against the Kiwis in Manchester.Five years ago they cruised through to the final and so far history has repeated itself here - the last try they conceded was scored by Englands Josh Charnley in the opening game of the tournament.That could be classed as a weakness, a te