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1、2012-2013学年第 二 学期常微分方程考试 AB卷答案 理学 院 年级 信息与计算科学 专业填空题（每题4分，共20分）1. 形如 （连续）的方程是 一阶线性微分 方程，它的通解为 .2. 形如的方程是 3 阶_齐次_(“齐次”还是”非齐次”)_常_系数的微分方程,它的特征方程为.3. 形如的方程为 欧拉 方程, 可通过变换把它转化成常系数方程.4. 满足初始条件：=0, =1的特解55.微分方程的解存在且唯一的条件是: 在R上连续且满足利普希茨条件 一、 下列微分方程的解（每题5分，共30分）1=解：令x+y=u,则=-1 .3 -1= u-arctgu=x+c y-arctg(x+y
2、)=c. .52解：两边同乘以得： .3故方程的通解为： .53解：令，则，两边对x求导，得 ， .3解之得 ，所以， .4且y=x+1也是方程的解，但不是奇解. .54. 解：特征方程有三重根， .3故通解为 .55. 解：特征方程有根0,齐线性方程的通解为x= .3又因为0是特征根，故可以取特解行如代入原方程解得A=，B= .4故通解为x= .56. 解: 原方程可化为 1分离变量可得 .3 两边积分可得 .4 将初值代入上式求得方程的解: .5二、 求下列方程（组）的通解（每题10分，共30分）1求一曲线，使其任一点的切线在轴上的截距等于该切线的斜率.解： 设为所求曲线上的任一点，则在点
3、的切线在轴上的截距为： .3由题意得 即 也即 两边同除以，得 .5即 .7即 .10为方程的解。2. 满足初值条件解:方程组的特征值, .2对应特征值的特征向量应满足 对任意常数, , 取, 得 .4对应特征值的特征向量应满足 对任意常数, , 取, 得 .6所以基解矩阵为: .8= .103.求方程 通过点 的第二次近似解.解: 令，于是 .5 .10五、应用题（10分）33. 摩托艇以5米/秒的速度在静水运动，全速时停止了发动机，过了20秒钟后，艇的速度减至米/秒。确定发动机停止2分钟后艇的速度。假定水的阻力与艇的运动速度成正比例。解：，又，由此 即 .5其中，解之得 又时，；时，。故得
4、 ，从而方程可化为 .7当时，有 米/秒 .8即为所求的确定发动机停止2分钟后艇的速度。 .10六、证明题 （10分）1、试证: 非齐次线性微分方程组的叠加原理:即: 设分别是方程组的解，则是方程组的解.证明： （1） （2）分别将代入（1）和（2）则 .5则令 即证 .102010-2011 学年第 二 学期常微分方程考试 B卷答案 理学 院 年级 信息与计算科学 专业一、 填空题（每题4分，共20分）1. 是恰当方程的充要条件是；其通解可用曲线积分表示为 .2方程 叫 克莱罗 方程，其通解是，其奇解是 .3. 形如的方程是 2 阶 非齐次 (“齐次”还是”非齐次”)_常系数的微分方程,它的
5、特征方程的特征根为 .4. 若 是同一线性方程 的基解方阵，则它们间有关系 .55.微分方程的解存在且唯一的条件是: 在R上连续且满足利普希茨条件 二、下列微分方程的解（每题5分，共30分）1. 解： 令 .1则： 即 得到 故 即 .4 另外也是方程的解。 . .52. =解： y= () .3=-e()+c=c e- ()是原方程的解。 .53。 设 .3 .4 ， 解为 .54. 解：特征方程有复数根, .3故通解为 .55.解: 原方程可化为 故 .56. 解：特征方程有根-2,-4 .1故齐线性方程的通解为x= .3-2是特征方程的根，故代入原方程解得A= .4故通解为x= .5三、
6、求下列方程（组）的通解（每题10分，共30分）1.解：特征方程有2重根-a.2当a=-1时，齐线性方程的通解为s=,1是特征方程的2重根，故代入原方程解得A=通解为s=,.6当a-1时，齐线性方程的通解为s=,1不是特征方程的根，故代入原方程解得A=故通解为s=+.102. 求其基解矩阵.解: det（EA）=0得， .3对应于的特征向量为u， （ 0 ）对应于的特征向量为v， （ ） .5u，v是对应于，的两个线性无关的特征向量(t)=是一个基解矩阵 .103. 求方程 通过点 的第二次近似解.解: 令，于是 .5 .10五、应用题（10分）1求一曲线，过点(1,1), 其任一点的切线在轴上
7、的截距等于.解： 设为所求曲线上的任一点，则在点的切线在轴上的截距为： .3由题意得 两边同除以，得 .5即 .7即 .8将代入上式得。.10六、证明题 （10分）1、 试证：如果是=Ax满足初始条件的解，那么expA(t-t)证明：由于(t)-1(t0) (t) .5又因为(t)= expAt , -1(t0)=( expAt0)-1= exp(-At0), f(s)=0,又因为矩阵 （At）（- At0）=（- At0）（At） .7所以 expA(t-t) .10Visa-free policy brings Chengdu biz, tourism boost. Making nati
8、onal headlines several times, Chengdus 72-hour visa-free policy has attracted wide attention from both Chinese and foreign experts and businessmen since it took effect on Sept 1 last year. The program permits citizens from 51 countries and regions including the United States, Australia, Canada and J
9、apan who have valid visas and flight tickets to a third country to spend three days in the city. The capital of Sichuan province is the first city in the western region of China to offer foreign tourists a three-day visa and the fourth nationwide to adopt the policy following Shanghai, Beijing and G
10、uangzhou. Li Zhiyong, deputy dean of the tourism institute at Sichuan University, said the move contributes to a large increase in the number of overseas tourists and raises the citys level of internationalization. The policy will also bring direct economic revenue, Li said. Chengdu has many cultura
11、l legacies and is also a paradise for panda lovers with the worlds largest breeding and research center. Three days are long enough for foreign visitors to visit those iconic tourist spots, he noted. The city is home to the remains of the Jin sha civilization that dates back more than 3,000 years as
12、 well as the Qing cheng Mountains and the Du jiang yan irrigation system. Qing cheng has long been recognized as the birthplace of Taoism, Chinas ancient indigenous religion, while Du jiang yan is considered to be the oldest functioning water-control project in the world. Chengdu ranked third in tou
13、rist facilities, management and services among 60 Chinese cities in a customer satisfaction survey released last year. But, Li added that efforts are still needed to develop more tourism products, improve English services and provide accurate translation of traffic signs and scenic billboards. Zhao
14、Yun, chairwoman of British Chamber of Commerce Southwest China, told China Daily that his colleagues found the policy very convenient. A British client once flew here and stayed for just one day to check her ordered goods, she said. Zhao was born in Shanxi province, but she has lived in Chengdu for
15、more than 10 years. My life was like a running race moving from place to place. I also lived in Beijing and Shanghai before, she said. But Chengdu is a place that you never want to leave once settling down. It is now my second hometown, she said. If the environment is further improved, the city will
16、 attract more people to visit and live, with the 72-hour visa-free policy and compelling conditions in transportation, culture, climate and cuisine, he said. Foreigners also gave positive feedback on the policy. A spokesman from Dell Inc said the company has a global hub of operation in Chengdu, so
17、the three-day visa has an immediate and positive influence on the companys business development. Rudy Buttignol, president of the public broadcasting company in British Columbia, Canada, said his work requires frequent travel to Chengdu and the policy makes the trips easier. Data from the citys publ
18、ic security bureau shows some 100 foreign visitors enjoyed the 72-hour policy by the end of March, most of them from the United States, the United Kingdom and Germany. Chengdu also reported robust growth in its overall tourist industry last year. Official statistics show that it received some 150 mi
19、llion tourists last year, an increase of 28 percent from 2012. Around 1.7 million came from abroad, an increase of 12 percent. Total revenue from tourism surpassed 133 billion yuan ($21.7billion). During his visit to Kazakhstan in September, Chinese President Xi Jinping proposed that China and Centr
20、al Asia join hands to build a Silk Road economic belt to boost cooperation. The idea has been widely echoed in Central Asian countries, becoming an encouraging blueprint for Chinese areas along the Silk Road that has linked Asia and Europe for more than 2,000 years. In the next three weeks, China Da
21、ily reporters will travel through the belt in China and in Kazakhstan, Uzbekistan and Turkey. They will show the progress and expectations of the countries, businesses and peoples on the route. Shaanxi - the start of the ancient Silk Road - has positioned itself as the new starting point for the dev
22、elopment of the Silk Road Economic Belt, which will strengthen Chinas cooperation with Central Asian countries, a senior official said. Shaanxi Governor Lou Qinjian said the province is fresh, rich and unique, as it was when it anchored one end of the ancient Silk Road. It is the best option for acc
23、ommodating industrial transfers from East China or the world at large, he said on Wednesday in Xian.Lou held a joint interview with 27 media, including China Daily, the first in a series of interviews entitled Chinese Media Along the Silk Road. The interviews will be in Shaanxi and Gansu provinces and the Xinjiang Uygur autonomous region, as well as Kazakhstan, Uzbekistan and Turkey.The media group held the first interview on Wednesday morning in Xian, the starting point of the ancient Silk Road, a trade channel established more than 2,000 years ago linking China, Central Asia and Europe.11