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1、常微分方程模拟试题一、填空题（每小题3分，本题共15分） 1一阶微分方程的通解的图像是 2 维空间上的一族曲线 2二阶线性齐次微分方程的两个解为方程的基本解组充分必要条件是 3方程的基本解组是 4一个不可延展解的存在在区间一定是 区间 5方程的常数解是 二、单项选择题（每小题3分，本题共15分） 6方程满足初值问题解存在且唯一定理条件的区域是（ ）（A）上半平面 （B）xoy平面 （C）下半平面 （D）除y轴外的全平面 7. 方程（ ）奇解（A）有一个 （B）有两个 （C）无 （D）有无数个 8连续可微是保证方程解存在且唯一的（ ）条件 （A）必要 （B）充分 （C）充分必要 （D）必要非充分
2、 9二阶线性非齐次微分方程的所有解（ ） （A）构成一个2维线性空间 （B）构成一个3维线性空间 （C）不能构成一个线性空间 （D）构成一个无限维线性空间 10方程过点(0, 0)有（ B）(A) 无数个解(B) 只有一个解 (C) 只有两个解(D) 只有三个解三、计算题（每小题分，本题共30分） 求下列方程的通解或通积分： 11. 12. 13. 14 15四、计算题（每小题10分，本题共20分） 16求方程的通解 17求下列方程组的通解 五、证明题（每小题10分，本题共20分） 18设在上连续，且，求证：方程的一切解，均有 19在方程中，在上连续，求证：若恒不为零，则该方程的任一基本解组的
3、朗斯基行列式是上的严格单调函数常微分方程模拟试题参考答案 一、填空题（每小题3分，本题共15分） 12 2线性无关（或：它们的朗斯基行列式不等于零）3 4开 5 二、单项选择题（每小题3分，本题共15分） 6D 7C 8B 9C 10A 三、计算题（每小题分，本题共30分）11解： 为常数解 （1分） 当，时，分离变量取不定积分，得 （3分） 通积分为 （6分）注：包含在常数解中，当时就是常数解，因此常数解可以不专门列出。 13解： 方程两端同乘以，得 （1分） 令 ，则，代入上式，得 （3分）这是一阶线形微分方程，对应一阶线形齐次方程的通解为 （4分）利用常数变易法可得到一阶线形微分方程的通
4、解为 （5分） 因此原方程通解为 （6分） 14解： 因为，所以原方程是全微分方程 （2分） 取，原方程的通积分为 （4分）计算得 （6分） 15解： 原方程是克莱洛方程，通解为 （6分） 四、计算题（每小题10分，本题共20分）16解： 对应齐次方程的特征方程为， （1分）特征根为， （2分）齐次方程的通解为 （4分） 因为是特征根。所以，设非齐次方程的特解为 （6分） 代入原方程，比较系数确定出 ， （9分） 原方程的通解为 （10分）17解： 齐次方程的特征方程为 （1分）特征根为 （2分）求得特征向量为 （3分）因此齐次方程的通解为 （4分） 令非齐次方程特解为 （5分）满足 （6分）
5、解得 （8分）积分，得 ， （9分）通解为 （10分） 五、证明题（每小题10分，本题共20分） 18证明： 设是方程任一解，满足，该解的表达式为 （4分） 取极限 = （10分） 19证明： 设，是方程的基本解组，则对任意，它们朗斯基行列式在上有定义，且又由刘维尔公式 ， （5分） 由于，于是对一切，有 或 故 是上的严格单调函数 （10分） Visa-free policy brings Chengdu biz, tourism boost. Making national headlines several times, Chengdus 72-hour visa-free policy
6、 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 Japan who have valid visas and flight tickets to a third country
7、 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 Guangzhou. Li Zhiyong, deputy dean of the tourism institute at S
8、ichuan 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 cultural legacies and is also a paradise for panda lovers with the wor
9、lds 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 well as the Qing cheng Mountains and the Du jiang yan irrigati
10、on 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 tourist facilities, management and services among 60 Chinese citie
11、s 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 Yun, chairwoman of British Chamber of Commerce Southwest China,
12、 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 more than 10 years. My life was like a running race moving from
13、 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 attract more people to visit and live, with the 72-hour visa-f
14、ree 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 the three-day visa has an immediate and positive influence on t
15、he 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 public security bureau shows some 100 foreign visitors enjoyed the
16、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 million tourists last year, an increase of 28 percent from 2012.
17、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 Central Asia join hands to build a Silk Road economic belt to boost
18、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 Daily reporters will travel through the belt in China and in Kaza
19、khstan, 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 development of the Silk Road Economic Belt, which will strengthen
20、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 accommodating industrial transfers from East China or the world at
21、 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.5