Learn General Secretary on "two to learn a" strengthening "four Consciousnesses" important speech caused a strong reaction in the country. Time, watching "red treasure", the origin of building the party back to power, how to strengthen services for the masses, improve party cohesion, fighting to become the grass-roots party members and masses hot topic. Grass-roots party organizations "two" is to strengthen the service of party members and cadres, the pioneer spirit. Distribution of grass-roots party organizations in all walks of people, clothing, shelter, which belongs to the nerve endings of the party organization and comments reputation has a direct perception of the masses. Strengthen the party ahead of the "pedal" spirit; strengthen the party members and cadres "success does not have to be me" and "the first to bear hardships, the last to" service spirit to set the party's positive image among the people is important. Grass-roots party organizations "two" is to cleanse all people not happy not to see "stereotypes", establish the honest faithful, diligent faith for the people. No need to avoid mentioning that, some members of our party can not stand the "money," corrosion of temptation, thin, Xu Zhou, such abuse and corrupt bribery, malfeasance borers, and rats. Two, is to clean up, thin, Xu, Zhou's solution to restore the party's fresh and natural, solid and honest work style. Cleansing "take, eat, card," undesirable and behaviour, "cross, hard and cold, push" attitude. Grass-roots party organizations "two" is to strengthen the sense of ordinary party members, participating in consciousness, unity consciousness. For reasons known, members of grass-roots party branches less mobile, less resources, and the construction of party organizations have some lag. Two studies, is to focus on the grass-roots party branches "loose, soft, loose" problem, advance the party members and cadres, "a gang working", "Hong Kong report." Strong cleanup actions, style and rambling, presumptuous "unqualified" party members, pays special attention to party members and cadres "joining party of thought" problem. "Party building" is obtained in the long-term development of our party's historical experience accumulated. Two is our party under the new historical conditions, strengthen the party's construction of a new "rectification movement." Grass-roots party organizations should always catch the hard work, results-oriented. Two educational outcomes are long-term oriented and become an important impetus for the work. "Two" should have three kinds of consciousness "two" study and education, basic learning lies in the doing. Only the Constitution address the series of party rules, and do solid work, be qualified party members had a solid ideological basis. Only the "learning" and "do" real unity, to form a "learn-learn-do-do" the virtuous cycle, and ultimately achieve the fundamental objective of education. This requires that the Organizationlearning education, need three kinds of consciousness: one is to establish an integrated awareness. "Learning" and "do" what car isTwo-wheel, bird wings, need to go hand in hand, one end can be neglected. Communist theoretician and man. Only by closely combining theory and practice together in order to truly realize their value. "Learning" is the Foundation, the Foundation is not strong, shaking; " "Is the key to net to net thousands of accounts. "Two" education, "" lay the basis, going to "do" the key grip, so that the "learning" and "doing" back to standard, so that the majority of party members "learn" learning theory of nutrients, in the "doing" practice party's purposes. Second, to establish a sense of depth. "Learning" and "do" not Chu drawn, entirely different, but the organic unity of the whole. "Two" learning education, we need to explore integrating "learning" in "do", exhibit "do" in "Science". To avoid the "learning" into simple room instruction, "do" into a monotone for doing. Should exploration "learn" in the has "do", "do" in the has "learn" of education and practice of carrier, makes general grass-roots members can in "learn" in the has "do" of achievements sense, in "do" in the has "learn" of get sense, real makes party of theory brain into heart, put for people service concept outside of Yu shaped. Third, to adhere to long-term the awareness. Style construction on the road forever, "two" had to catch the long-term. "Two" study and education, by no means, assault-style wind-sport, but the recurrent education within the party. In recent years, the party's mass line education practice and "three-three" special education in grass-roots borne rich fruits, vast numbers of party members and cadres withstood the baptism of the spirit. "Two" greater need to focus on longer hold long-term, to establish and perfect the effective mechanism of the education, focusing on the creation of long-term education, strive to make the vast number of party members to maintain their vanguard Color, maintain the party's advanced nature and purity. Awareness-raising, antennas and atmosphere – a discussion on how leading cadres of party members "two" current, "two" activity is in full swing up and down the country, party cadres as a "key minority" is both a barometer and impetus. The "two" meaning enough deep, is to determine the party cadres can resolve to study hard first. In the "two" in the process, some cadres of himself, standing long, high awareness, that Constitution Party rules is simple, its not worth bothering some party cadres think speak series has nothing to do with the grass-roots work, water business learning series of speeches seen as window dressing. These "lazy, casual, and decadent" ideas learning lacks motivation, a serious impediment to "two" effect. John Stuart Mill once said, only a basic element of human thought patterns change dramatically, human destiny can make great improvement. The same, only party members and2.2 一阶线性方程与常数变易公式(First order linear differential equationand constant variation formula )[教学内容] 1. 认识一阶线性齐次方程和一阶线性非齐次方程; 2.介绍一阶线性非齐次方程的常数变易公式; 3. 介绍电学知识和基尔霍夫定律; 4. 认识Bernoulli 方程及其通过变量替换化为一阶线性方程的解法; 5. 介绍其他可化为一阶线性方程的例子.[教学重难点] 重点是知道一阶线性非齐次方程的解法，难点是如何根据方程的形式引入新的变量变换使得新方程为一阶线性方程.[教学方法] 自学1、4；讲授2、3 课堂练习 [考核目标]1. 熟练运用常数变易公式;2. 知道计算和一些三角函数恒等式; 3. 知道电⎰dx bx sin e ax 学一些知识，如电容电流公式、电感电压公式和基尔霍夫定律; 4. 知道溶液混合问题建模; 5. 认识Bernoulli 方程并会经过适当变换化为线性方程求解. 6. 知道交换自变量和因变量化非线性方程为一阶线性方程.1. 认识一阶线性齐次方程和一阶线性非齐次方程（First order (non)homogeneous linear differential equation ）(1) 称形如的方程为一阶线性齐次方程，其中连续；y p(x)dxdy=p(x)称形如的方程为一阶线性非齐次齐次方程，其中连续且q(x)y p(x)dxdy+=q(x) p(x),不恒为零.q(x)(2) 当时，改写为0y ≠y p(x)dxdy=，其中表示P(x)的一个原1C dx p(x)|y |ln ,dx p(x)y dy dx, p(x)y dy +===⎰⎰⎰⎰dx p(x)函数(antiderivative). 因此，通解(general solution)为，y p(x)dxdy =1C p(x)dx e C ~ ,eC ~y =⎰±=此外y=0也是解. 综上，的解为为任意常数.y p(x)dxdy =C ,eC y p(x)dx⎰=(3) 常数变易法：如何求的解呢？q(x)y p(x)dxdy+=假定上述线性非齐次方程有如下形式的解 ，则代入原方程来确定C(x)，⎰=p(x)dxeC(x)y ，q(x)p(x)C(x)e e p(x) C(x)e (x)' C dxdy p(x)dxp(x)dx p(x)dx +⎰=⎰+⎰=即，，此处C 为q(x)e(x)' C p(x)dx=⎰C q(x)dx eC(x) q(x), e(x)' C p(x)dx-p(x)dx+⎰=⎰=⎰-表示在时刻t 时电容两端电压，则根据电学知识，电容两端电量电阻两端电压为R I=. 由基尔霍夫定律知，dtdUR . 改写为 ，这是一个一阶线性非齐次方程0dt dU =RC EU RC 1dt dU +⋅-=ELearn General Secretary on "two to learn a" strengthening "four Consciousnesses" important speech caused a strong reaction in the country. Time, watching "red treasure", the origin of building the party back to power, how to strengthen services for the masses, improve party cohesion, fighting to become the grass-roots party members and masses hot topic. Grass-roots party organizations "two" is to strengthen the service of party members and cadres, the pioneer spirit. Distribution of grass-roots party organizations in all walks of people, clothing, shelter, which belongs to the nerve endings of the party organization and comments reputation has a direct perception of the masses. Strengthen the party ahead of the "pedal" spirit; strengthen the party members and cadres "success does not have to be me" and "the first to bear hardships, the last to" service spirit to set the party's positive image among the people is important. Grass-roots party organizations "two" is to cleanse all people not happy not to see "stereotypes", establish the honest faithful, diligent faith for the people. No need to avoid mentioning that, some members of our party can not stand the "money," corrosion of temptation, thin, Xu Zhou, such abuse and corrupt bribery, malfeasance borers, and rats. Two, is to clean up, thin, Xu, Zhou's solution to restore the party's fresh and natural, solid and honest work style. Cleansing "take, eat, card," undesirable and behaviour, "cross, hard and cold, push" attitude. Grass-roots party organizations "two" is to strengthen the sense of ordinary party members, participating in consciousness, unity consciousness. For reasons known, members of grass-roots party branches less mobile, less resources, and the construction of party organizations have some lag. Two studies, is to focus on the grass-roots party branches "loose, soft, loose" problem, advance the party members and cadres, "a gang working", "Hong Kong report." Strong cleanup actions, style and rambling, presumptuous "unqualified" party members, pays special attention to party members and cadres "joining party of thought" problem. "Party building" is obtained in the long-term development of our party's historical experience accumulated. Two is our party under the new historical conditions, strengthen the party's construction of a new "rectification movement." Grass-roots party organizations should always catch the hard work, results-oriented. Two educational outcomes are long-term oriented and become an important impetus for the work. "Two" should have three kinds of consciousness "two" study and education, basic learning lies in the doing. Only the Constitution address the series of party rules, and do solid work, be qualified party members had a solid ideological basis. Only the "learning" and "do" real unity, to form a "learn-learn-do-do" the virtuous cycle, and ultimately achieve the fundamental objective of education. This requires that the Organizationlearning education, need three kinds of consciousness: one is to establish an integrated awareness. "Learning" and "do" what car isTwo-wheel, bird wings, need to go hand in hand, one end can be neglected. Communist theoretician and man. Only by closely combining theory and practice together in order to truly realize their value. "Learning" is the Foundation, the Foundation is not strong, shaking; " "Is the key to net to net thousands of accounts. "Two" education, "" lay the basis, going to "do" the key grip, so that the "learning" and "doing" back to standard, so that the majority of party members "learn" learning theory of nutrients, in the "doing" practice party's purposes. Second, to establish a sense of depth. "Learning" and "do" not Chu drawn, entirely different, but the organic unity of the whole. "Two" learning education, we need to explore integrating "learning" in "do", exhibit "do" in "Science". To avoid the "learning" into simple room instruction, "do" into a monotone for doing. Should exploration "learn" in the has "do", "do" in the has "learn" of education and practice of carrier, makes general grass-roots members can in "learn" in the has "do" of achievements sense, in "do" in the has "learn" of get sense, real makes party of theory brain into heart, put for people service concept outside of Yu shaped. Third, to adhere to long-term the awareness. Style construction on the road forever, "two" had to catch the long-term. "Two" study and education, by no means, assault-style wind-sport, but the recurrent education within the party. In recent years, the party's mass line education practice and "three-three" special education in grass-roots borne rich fruits, vast numbers of party members and cadres withstood the baptism of the spirit. "Two" greater need to focus on longer hold long-term, to establish and perfect the effective mechanism of the education, focusing on the creation of long-term education, strive to make the vast number of party members to maintain their vanguard Color, maintain the party's advanced nature and purity. Awareness-raising, antennas and atmosphere – a discussion on how leading cadres of party members "two" current, "two" activity is in full swing up and down the country, party cadres as a "key minority" is both a barometer and impetus. The "two" meaning enough deep, is to determine the party cadres can resolve to study hard first. In the "two" in the process, some cadres of himself, standing long, high awareness, that Constitution Party rules is simple, its not worth bothering some party cadres think speak series has nothing to do with the grass-roots work, water business learning series of speeches seen as window dressing. These "lazy, casual, and decadent" ideas learning lacks motivation, a serious impediment to "two" effect. John Stuart Mill once said, only a basic element of human thought patterns change dramatically, human destiny can make great improvement. The same, only party members and解：设I(t)表示时刻t 时电感线圈上电流强度，则由电学知识有，电感线圈两端电压为. 由基尔霍夫定律知，闭合回路电压降为零. 于是 . 改写为dt dIL 0dt dIL I R E =--, 这是一个一阶线性非齐次方程. sin wt U L1L I R dt dI m +-=记, 由常数变易公式得到，wt sin L Uq(t) ,L R p(t)m =-=)C ~dt sin wt LU e (e )C ~q(t)dt e(eI(t)m L RtL Rt p(t)dtp(t)dt⎰⎰+=+⎰⎰=--.ba btcos b bt sin a e bt))isin bt (cos e b a ib)(a Im()e ib a 1Im()dt e Im(dt )Im(e e dt bt sin e 22ata22ib)t (a ib)t (a ibt at at +-=+⋅+-=+===++⎰⎰⎰22t LR mLRt m m LRtw (R/L) wt)cos w sin wt LR(e LU dt sin wt e LUdt sin wt L U e+-==⎰⎰令，2222w(R/L)wφsin ,w(R/L)R/L φ cos +-=+=于是由知，B sin A cos B cos A sin B)sin(A +=+，于是.22t LR m m LRtw (R/L)φ)sin(wt e L U dt sin wt L U e++=⎰L Rt 22m e C ~w (R/L)φ)sin(wt LU I(t)-+++=再注意到初始条件I(0)=0，，因此，22m0022m w(R/L)φsin L U C ~0,C ~e e w (R/L)φsin LU I(0)+-==++=.t LR22m22m ew(R/L)sin(φLUw (R/L)φ)sin(wt L U I(t)-+-++=)练习23. (1) 求； (2) 改写为，给出dt bt cos e at ⎰t cos b sin t a +θ)sin(t ba 122++所满足的条件. (3) 由 Euler 公式和推导出:θb sin i b cos e ib +=R b a, ,e e e b)i(a b i a i ∈=⋅+和b asin sin b cos a cos b)cos(a b,sin a cos b cos a sin b)sin(a -=++=+learning education, need three kinds of consciousness: one is to establish an integrated awareness. "Learning" and "do" what car isTwo-wheel, bird wings, need to go hand in hand, one end can be neglected. Communist theoretician and man. Only by closely combining theory and practice together in order to truly realize their value. "Learning" is the Foundation, the Foundation is not strong, shaking; " "Is the key to net to net thousands of accounts. "Two" education, "" lay the basis, going to "do" the key grip, so that the "learning" and "doing" back to standard, so that the majority of party members "learn" learning theory of nutrients, in the "doing" practice party's purposes. Second, to establish a sense of depth. "Learning" and "do" not Chu drawn, entirely different, but the organic unity of the whole. "Two" learning education, we need to explore integrating "learning" in "do", exhibit "do" in "Science". To avoid the "learning" into simple room instruction, "do" into a monotone fordoing. Should exploration "learn" in the has "do", "do" in the has "learn" of education and practice of carrier, makes general grass-roots members can in "learn" in the has "do" of achievements sense, in "do" in the has "learn" of get sense, real makes party of theory brain into heart, put for people service concept outside of Yu shaped. Third, to adhere to long-term the awareness. Style construction on the road forever, "two" had to catch the long-term. "Two" study and education, by no means, assault-style wind-sport, but the recurrent education within the party. In recent years, the party's mass line education practice and "three-three" special education in grass-roots borne rich fruits, vast numbers of party members and cadres withstood the baptism of the spirit. "Two" greater need to focus on longer hold long-term, to establish and perfect the effective mechanism of the education, focusing on the creation of long-term education, strive to make the vast number of party members to maintain their vanguard Color, maintain the party's advanced nature and purity. Awareness-raising, antennas and atmosphere – a discussion on how leading cadres of party members "two" current, "two" activity is in full swing up and down the country, party cadres as a "key minority" is both a barometer and impetus. The "two" meaning enough deep, is to determine the party cadres can resolve to study hard first. In the "two" in the process, some cadres of himself, standing long, high awareness, that Constitution Party rules is simple, its not worth bothering some party cadres think speak series has nothing to do with the grass-roots work, water business learning series of speeches seen as window dressing. These "lazy, casual, and decadent" ideas learning lacks motivation, a serious impediment to "two" effect. John Stuart Mill once said, only a basic element of human thought patterns change dramatically, human destiny can make great improvement. The same, only party members and3. Bernoulli 方程及其解法称形如为Bernoulli 方程. R n ,y q(x)y p(x)dxdyn ∈+=解法：当时，改写原方程,0y ≠1n , n)q(x)(1y p(x) n)(1dxdy y n)-(1n -1n -≠-+-=令，这是一个一阶线性非齐次方程.n)q(x)(1n)p(x)u (1dx du ,y u n1-+-==-例31 求解方程.2y x xy6dx dy -=解：经过观察，原方程是一个Bernoulli 方程, n=2. （1）当时，改写原方程为，令，则0y ≠ x 2)(1y x62)(1dx dy 2)y-(1212---=--21y u -=. 由常数变易公式得到， x u x6dx du +-=.6276-dx x6dx x6xC8x C)dx x (x )C xdx e(eu(x)+=+=+⎰⎰=⎰⎰-返回原变量得到.62xC8x y 1+=(2) 当y=0时，容易验证也是原方程的解. 0y =作业26. 求解方程（1）； （2）. 33y x y x dxdy=+1y(1) ,y xy 'y x 22==-4. 交换自变量和因变量化非线性方程为一阶线性方程例32. 求解（1）； （2）. 2y 2x y dx dy -=33yx xy 1dx dy -=解：(1) 这是一个一阶方程，非线性方程，不是Bernoulli 方程.(a) 当时，交换自变量和因变量而改写原方程为 . 这是一个0y ≠y x y2y y 2x dy dx 2-=-=一阶线性方程. 由常数变易公式得到， ，C)y)dy (e(ex dy y2dy y2+-⎰⎰=⎰-即 为所求方程的通积分.|)y |ln (C y C)y)dy (y1(y x 222-=+-=⎰(b) 当y=0时，已验证y=0也是原方程的一个解. (2) 结合Bernoulli 方程来完成，留作练习.作业27. 求解方程（1）； （2） . 3y x ydx dy +=y 2y x dx dy 22+=learning education, need three kinds of consciousness: one is to establish an integrated awareness. "Learning" and "do" what car isTwo-wheel, bird wings, need to go hand in hand, one end can be neglected. Communist theoretician and man. Only by closely combining theory and practice together in order to truly realize their value. "Learning" is the Foundation, the Foundation is not strong, shaking; " "Is the key to net to net thousands of accounts. "Two" education, "" lay the basis, going to "do" the key grip, so that the "learning" and "doing" back to standard, so that the majority of party members "learn" learning theory of nutrients, in the "doing" practice party's purposes. Second, to establish a sense of depth. "Learning" and "do" not Chu drawn, entirely different, but the organic unity of the whole. "Two" learning education, we need to explore integrating "learning" in "do", exhibit "do" in "Science". To avoid the "learning" into simple room instruction, "do" into a monotone for doing. Should exploration "learn" in the has "do", "do" in the has "learn" of education and practice of carrier, makes general grass-roots members can in "learn" in the has "do" of achievements sense, in "do" in the has "learn" of get sense, real makes party of theory brain into heart, put for people service concept outside of Yu shaped. Third, to adhere to long-term the awareness. Style construction on the road forever, "two" had to catch the long-term. "Two" study and education, by no means, assault-style wind-sport, but the recurrent education within the party. In recent years, the party's mass line education practice and "three-three" special education in grass-roots borne rich fruits, vast numbers of party members and cadres withstood the baptism of the spirit. "Two" greater need to focus on longer hold long-term, to establish and perfect the effective mechanism of the education, focusing on the creation of long-term education, strive to make the vast number of party members to maintain their vanguard Color, maintain the party's advanced nature and purity. Awareness-raising, antennas and atmosphere – a discussion on how leading cadres of party members "two" current, "two" activity is in full swing up and down the country, party cadres as a "key minority" is both a barometer and impetus. The "two" meaning enough deep, is to determine the party cadres can resolve to study hard first. In the "two" in the process, some cadres of himself, standing long, high awareness, that Constitution Party rules is simple, its not worth bothering some party cadres think speak series has nothing to do with the grass-roots work, water business learning series of speeches seen as window dressing. These "lazy, casual, and decadent" ideas learning lacks motivation, a serious impediment to "two" effect. John Stuart Mill once said, only a basic element of human thought patterns change dramatically, human destiny can make great improvement. The same, only party members and5. 一些一阶线性方程的理论（1）考虑方程，其中p(x), q(x)都是以w>0为周期的连续函数. 用常数q(x)y p(x)dxdy=+变易公式证明：（a) 若，则方程任一非零解都以w 为周期的周期函数充要条件是0q(x)≡p(x)的平均值 (b) 若不恒为零，则方程有唯一w 周期解充要.0p(x)dx w 1(x)p w==⎰q(x)条件是, 试求出此解. （参见丁同仁、李承治《常微分方程教程》0p(x)dx w 1(x)p w≠=⎰P36 习题5， 6）。
