浅析常数变易法PPT课件
常数变易法
常数变易法
常数变易法是求解复杂问题中经常采用的一种方法，它既可以帮助我们求解复杂问题，又可以帮助我们节省时间，提高效率。
但是，要想有效地使用常数变易法，我们需要对它有全面的认识和理解，并能够熟练掌握运用它的相关技巧。
首先，我们来了解它的定义，常数变易法就是从现有的函数中求解函数变形的方法，它的关键就是利用函数的变易性，将原始的函数变形为一个简单的函数，让求解问题更加容易。
例如，如果我们要求解一个立方函数，我们可以利用常数变易法，将其变形为一个平方函数，这样就可以用更简单的方式来求解。
其次，在掌握常数变易法的时候，我们需要学习它的基本原理，主要是利用二次函数的“常数变易”原理，即一次函数可以表示为一次函数与常数相乘的形式。
换句话说，利用“常数变易”原理，我们可以将复杂的函数变形为更为简单的函数，从而求解复杂的函数。
此外，为了有效地运用常数变易法，我们还需要掌握一些算法，才能够更加高效地求解复杂函数。
比如，我们可以用分治算法来求解复杂的函数，而且分治算法可以从另一个角度来分析函数，从而使函数的求解更加容易。
总的来说，常数变易法是一种解决复杂问题的高效方法，它可以帮助我们通过变易函数的方式节省时间，提高效率。
但是，如果要有效地使用常数变易法，我们还需要学习它的基本原理、熟练掌握它的算法，这样才能够有效地求解复杂的函数。
第4节 常数变易法
令 y∗( x) = Φ( x) c( x),
待定
(4.1)
待定 的解， 若 y∗( x)是(3.1)s的解，则 y∗′ ( x) ≡ A( x) y∗( x) + f ( x)
令 y∗( x) = Φ( x) c( x),
(4.1)
∵ y∗'( x) = Φ′( x) c( x) + Φ( x) c′( x)
由线性微分方程(组 的通解结构定理 的通解结构定理10′ 由线性微分方程 组)的通解结构定理 ′ (或定理 )，知求 或定理10 ，知求(3.1)s 的通解的关键： 的通解的关键 关键： 或定理 ① 确定与其相对应的齐线性方程(3.2)s 确定与其相对应的齐线性方程 的一个基本解组; 的一个基本解组 的一个特解: ② 求(3.1) 的一个特解 y *(x) .
而
Φ−1( x) =
Φ*( x) det Φ( x)
… Wn1 W W 11 21 1 W = ⋅ 12 W22 … Wn2 W( x) W n W2n … Wnn 1
…
y2( x) ⋯ yn( x) y1( x) 其中 W ( x)( x) ( x)的第 行，第(jx) ij y′ : Φ y′ ( x) i ⋯ y′ 列元素的 2 n Φ( x) = 1 ⋮ ⋯ ⋮ . 代数余子式⋮ (n−1) (n−1) (n−1) ( x) y1 ( x) y2 ( x) ⋯ yn
∫
x2 x x2 x e x + sinx e x sin x = e + sin x + e . = 2 ⋅ 2 + 0 1 1 1 0
x2 x y1 = e x + sin x + e 即 2 . y2 = 1 方法2. 方法 由(1)，知原方程组为 ， dy1 = y1 + (cos x − sin x) y2 + xe x dx dy2 = 0, dx y1(0) = y2(0) = 1
2.2线性微分方程与常数变易法
练习 解
(3)
dy y 3 dx x y
1 , Q( y ) y 2 y
方程可以改写为:
dx 1 x y2 dy y
故通解为:
p( y )
xe
即:
1 dy y
( y e
2
1 dy y
1 2 c) y ( y c) 2
1 3 x y cy 2
§ 2.2 Linear ODE and variation of constants Method
dc ( x) ( x 1) n nc ( x)( x 1) n 1 nc ( x)( x 1) n 1 e x ( x 1) n dx
x dc ( x ) x c ( x ) e c 积分得 e 即 dx n x y ( x 1) (e c), c 为任意常数 故通解为
2) 用公式求方程通解
2
xe
1 dx y
[ ye
2
1 dy y
dy c] e
ln y 2
( ye
ln y 2
dy c)
1 x y ( dy c) y 2 ln y cy2 y x y 2 ln y cy2
2
§ 2.2 Linear ODE and variation of constants Method
练习 (2)
解
dy 2 xy 4 x dx
用公式求解, p( x) 2 x, Q( x) 4 x
ye
e
2 xdx
x2
e
即:
x2
( 4 xe dx c)
线性微分方程的常数变易法
线性微分方程的常数变易法线性微分方程是微积分中重要的研究对象，常数变易法是解线性微分方程的一种常用方法。
本文将介绍线性微分方程以及常数变易法的基本概念和步骤。
1. 线性微分方程的定义和形式线性微分方程是指形如y'' + p(x)y' + q(x)y = r(x)的微分方程，其中p(x)、q(x)和r(x)为已知函数，y为未知函数。
一阶线性微分方程可以表示为y' + p(x)y = q(x)。
2. 常数变易法的基本思想常数变易法是对齐次线性微分方程的解进行求解的一种方法。
首先求得齐次线性微分方程的通解，然后利用常数变易法找出非齐次线性微分方程的一个特解，将通解和特解相加得到非齐次线性微分方程的通解。
3. 常数变易法的步骤步骤一：求齐次线性微分方程的通解对于齐次线性微分方程y'' + p(x)y' + q(x)y = 0，我们可以先求得其特征方程。
特征方程是通过将y替换为е^(rx)得到的方程，其中r为常数。
解特征方程可以得到一组线性无关的解，它们的线性组合就是齐次线性微分方程的通解。
步骤二：求非齐次线性微分方程的特解对于非齐次线性微分方程y'' + p(x)y' + q(x)y = r(x)，我们假设其特解为y = u(x)v(x)，其中u(x)为常数，v(x)为齐次线性微分方程的通解。
将特解y代入非齐次线性微分方程，可以得到一个关于u(x)的方程，若能解出u(x)的具体形式，则可以得到非齐次线性微分方程的一个特解。
步骤三：求非齐次线性微分方程的通解将齐次线性微分方程的通解和非齐次线性微分方程的特解相加，即可得到非齐次线性微分方程的通解。
4. 常数变易法的应用举例以一阶线性微分方程y' + p(x)y = q(x)为例，根据常数变易法的步骤，首先求得齐次线性微分方程y' + p(x)y = 0的通解，然后假设特解为y =u(x)v(x)，将特解代入非齐次线性微分方程，解出u(x)的具体形式，最后将通解和特解相加即可得到非齐次线性微分方程的通解。
用常数变易法求解二阶非齐次线性微分方程
其根
则①对应的齐次方程的通解为
第10页/共24页
设特解: y At 2 B t C 代入①确定系数, 得
① 的通解为 换回原变量, 得原方程通解为
第11页/共24页
例2. 解: 将方程化为 则方程化为
(欧拉方程)
即
②
特征根:
设特解: y At 2et , 代入 ② 解得 A = 1, 所求通解为
于是欧拉方程 xn y(n) p1 xn1 y(n1) pn1 x y pn y f ( x)
转化为常系数线性方程:
Dn y b1Dn1 y bn y f (et )
即
dn dt
y
n
b1
dn1 y d t n1
bn
y
f (原方程化为
亦即
①
特征方程
C1 cos(2ln
x)
C2
sin( 2 ln
x)
1 x
利用初始条件④得
C1 1,
C2
1 2
故所求特解为
y cos(2ln x) 1 sin(2ln x) 1
2
x
hw：p319 2,4.
第14页/共24页
一类特殊变系数非齐次线性微分方程
Euler Equation：
xn y(n) a1 xn1 y(n1) an1 xy an y f ( x)
1 x2
d2 y dt2
dy dt
计算繁!
x2
y
d2 dt
y
2
d d
y t
第8页/共24页
记D d , dt
Dk
dk dtk
(k 2, 3, ),则由上述计算可知:
x y D y
64常数变易法
此外, 他对
双纽线, 悬链线和对数螺线都有深入的研究 .
内容小结
1. 一阶线性方程
方法1 先解齐次方程 , 再用常数变易法.
方法2 用通解公式
化为线性方程求解.
2. 伯努利方程
思考与练习
判别下列方程类型:
提示:
可分离 变量方程
齐次方程
线性方程
线性方程
伯努利方程
习题6.1 3; 习题6.2 1 (1); 2 (3); 3 (2); 4(2)
一、一阶线性微分方程
一阶线性微分方程标准形式:
若 Q(x) 0,
若 Q(x) 0,
称为非齐次方程 .
1. 解齐次方程
分离变量
两边积分得
故通解为
称为齐次方程 ;
对应齐次方程通解
齐次方程通解
非齐次方程特解
2. 解非齐次方程
用常数变易法:
则
故原方程的通解
即
即
作变换
两端积分得
例1. 解方程
解: 先解
书中给出的伯努利数在很多地方有用,
伯努利(1654 – 1705)
瑞士数学家,
位数学家.
标和极坐标下的曲率半径公式,
1695年
版了他的巨著《猜度术》,
上的一件大事,
而伯努利定理则是大数定律的最早形式.
年提出了著名的伯努利方程,
他家祖孙三代出过十多
1694年他首次给出了直角坐
1713年出
作业
备用题
1. 求一连续可导函数
使其满足下列方程:
提示:
令
则有
利用公式可求出
2. 设有微分方程
其中
欧拉方程和常数变异法
则有
xy dy Dy, dt
x
2
y
d2y dt 2
dy dt
(
d2 dt 2
d dt
)
y
(
D2
D)
y
D(
D1)
y,
x
3
y
d3y dt 3
3 dd2t 2y
2 dy dt
(
D3
3D2
2D)
y
D( D1)(D
2)
y,
一般地，有 xk y(k ) D(D 1)(0D 2) (D k 1) y ，
令 xet , 化为 y 关于 t 的二阶常系数线性微分方程，
利用算子 D d ， xy Dy ， x2 y D(D1) y ， dt
把原方程写成：D(D1) ya1Dya2 y f (et ) ,
化简得 [D2 (a1 1)Da2 ]y f (et ) ，
即
d2y dt 2
(a1
1) dy dt
a2
d2y dx2
1 x
d dt
(
dy dt
) dt dx
1 x2
dy dt
1 x2
(
d2y dt 2
dy dt
)
，
d3 dx
y
3
2 x3
(
d2y dt 2
dy dt
)
1 x2
(
1 x
d3y dt 3
1 x
d2y dt 2
)
1 x3
(
d3y dt 3
d2y 3 dt2
2
dy dt
)
利用算子 D d ， dt
2
A 2 A A1
非齐次线性方程通解求法------常数变易法共37页PPT
16、人民应该为法律而战斗，就像为 了城墙 而战斗 一样。 ——赫 拉克利 特 17、人类对于不公正的行为加以指责 ，并非 因为他 们愿意 做出这 种行为 ，而是 惟恐自 己会成 为这种 行为的 牺牲者 。—— 柏拉图 18、制定法律法令，就是为了不让强 者做什 么事都 横行霸 道。— —奥维 德 19、法律是社会的习惯和思想的结晶 。—— 托·伍·威尔逊 20、人们嘴上挂着的法律，其真实含 义异常珍贵的东西，从任何源泉吸 收都不可耻。——阿卜·日·法拉兹
42、只有在人群中间，才能认识自 己。——德国
43、重复别人所说的话，只需要教育； 而要挑战别人所说的话，则需要头脑。—— 玛丽·佩蒂博恩·普尔
44、卓越的人一大优点是：在不利与艰 难的遭遇里百折不饶。——贝多芬
45、自己的饭量自己知道。——苏联
常数变易法
dx x 1
y x 1
积分得
即 y C(x 1)2
用常数变易法求特解. 令 y u (x) (x 1)2 , 则
y u (x 1)2 2u (x 1)
代入非齐次方程得 解得 故原方程通解为
u
2
(x
3
1) 2
C
3
22
例2. 解方程
方法二(公式法解):
把物体降落的铅垂线取作s轴，其指向朝下（朝向地心）。 设物体在时刻t的位置为s＝s(t)。物体受重力 F=mg的作用而自由下落，物体下落运动的加速度
a
d 2s dt 2 .
由牛顿第二定律F=ma，得物体在下落过
程中满足的关系式为
m
d 2s dt 2
mg,
或
d 2s dt 2
g.
4
微分方程的基本概念
b M
,
M max f (x, y) . ( x, y)D
9
§5.2.2 解的存在与唯一性定理
可分离变量方程
dy dx
f1 ( x)
f2 ( y)
M1(x)M 2 ( y) dx N1(x) N2 ( y) dy 0
解分离变量方程 g( y)dy f (x)dx
10
可分离变量的微分方程
y0，
如果 f (x, y) 在矩形区域 D: x x0 a, y y0 b内连续，
而且对于 y适合利普希茨条件
f (x, y1) f (x, y2 ) L y1 y1
则初值问题在区间I [x0 h, x0 h] 上存在唯一解， 其中常数
h
常微分方程-常数变异法.ppt
即
dc(x) ex dx
积分得
~
c(x) ex c
~
~
故通解为 y (x 1)n (ex c), c为任意常数
例2 求方程
dy y dx 2x y2 通解.
解: 原方程不是未知函数 y的线性方程 ，但将它改写为
dx 2x y2
dy y
即
dx 2 x y dy y
它是以x为未知函数 , y为自变量的线性方程 ，
的通解.
解: 这是Bernoulli 方程, n1 z x2 dx x
解以上线性方程得
z
e
1 x
dx
(
x
2e
1 x
dx
dx
c)
cx 1 x3 2
将z y2代入得所给方程的通解 为:
y2 cx 1 x3 2
二 线性微分方程的应用举例
例5 R-L串联电路.,由电感L,电阻R和电源所组成的串联电 路,如图所示,其中电感L,电阻R和电源的电动势E均为常数, 试求当开关K合上后,电路中电流强度I与时间t之间的关系.
电路的Kirchhoff第二定律: 在闭合回路中,所有支路上的电压的代数和为零.
解: 设当开关K合上后, 电路中在时刻t的电流强度为I(t), 则电流经过电感L, 电阻R的电压降分别为L dI , RI,
dt
于是由Kirchhoff第二定律, 得到
L dI RI E. dt
取开关闭合时的时刻为0, 即I (0) 0.
dx x 1
从
dy n y 分离变量得 dx x 1
dy n dx y x 1
两边积分得 ln y nln x 1 c1
故对应齐次方程通解为 y c(x 1)n
常数变易法的原理
常数变易法的原理
常数变易法是一种数学方法，用于求解特定类型的问题。
它的原理是通过假设一个未知数为常数，并在后续计算中逐步调整这个常数，以便解决问题。
使用常数变易法的关键是找到一个适当的常数，使得问题的解可以用这个常数来表示。
一般来说，常数经过调整后可以使问题简化，或者使得解的形式更加容易处理。
在使用常数变易法时，首先需要假设一个常数，并将其视为未知数，然后将这个常数代入问题的表达式或方程中进行计算。
根据计算结果，通过适当调整常数的值，逐步逼近或找到问题的解。
常数变易法的思想通常用于求解微积分、微分方程和变分问题等数学领域中的一些特殊问题。
它的目的是通过假设一个常数来简化问题，使得求解变得更加容易和直观。
总之，常数变易法是一种使用常数作为未知数，并通过逐步调整常数的值来求解问题的方法。
它可以简化问题，使得计算更加方便，从而得到问题的解。
3-7-一阶线性方程与常数变易法
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）。
3-7-一阶线性方程与常数变易法精品文档5页
2.2 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⎰dx bx sin e ax 计算和一些三角函数恒等式; 3. 知道电学一些知识，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y p(x)dxdy=的方程为一阶线性齐次方程，其中p(x)连续； 称形如q(x)y p(x)dxdy+=的方程为一阶线性非齐次齐次方程，其中q(x) p(x),连续且q(x)不恒为零. (2) 当0y ≠时，改写y p(x)dxdy=为 1C dx p(x )|y |ln ,dx p(x )y dy dx , p(x )y dy +===⎰⎰⎰，其中⎰dx p(x)表示P(x)的一个原函数(antiderivative). 因此，y p(x)dxdy=通解(general solution)为1C p(x)dx e C ~ ,eC ~y =⎰±=，此外y=0也是解. 综上，y p(x)dxdy =的解为C ,eC y p(x)dx⎰=为任意常数. (3) 常数变易法：如何求q(x)y p(x)dxdy+=的解呢？ 假定上述线性非齐次方程有如下形式的解 ⎰=p(x)dxeC(x )y ，则代入原方程来确定C(x)，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+⎰=⎰=⎰-，此处C 为任意常数,⎰⎰q(x )dx ep(x)dx-为函数q(x )ep(x)dx-⎰一个原函数.综上，一阶线性非齐次方程的通解为⎰⎰⎰⎰+⎰=+⎰⋅⎰=q(x )dx eeCeC)q(x )dx e(ey(x )p(x)dx-p(x)dxp(x)dxp(x)dx-p(x)dx.2. 一些实际应用例子（Applications ） 例28. 电容器的充电和放电模型RC 电路：假定开始电容C 上没有电荷，电容两端电压为0，合上开关1后，电池E 对电容C 开始充电，电池电压为E ，电阻阻值为R ，电容C 两端电压逐渐上升. 写出充电过程中，电容C 两端电压随时间变化的规律.解：设U(t)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Q=U C ，电流I =dt dU C dt dQ =, 电阻两端电压为R I=dtdUR . 由基尔霍夫定律知，闭合回路上压降为零. 即有0dt dU RC U E =--. 改写为 RC EU RC 1dt dU +⋅-=，这是一个一阶线性非齐次方程. 记RCEq(t) ,RC 1p(t)=-=, 由常数变易公式得到， C~e E )C ~(Ee e )C ~dt RCE e (e )C ~q(t)dt e(eU(t)RC tRC t RC t RC t RC t p(t)dtp(t)dt----+=+=+=+⎰⎰=⎰⎰再注意到初始条件U(0)=0，-E C ~0,C ~e Ee U(0)0==+=，因此，RC tEe E U(t)--=.例29. 考察如下RL 电路图，设电源E 的电压为0 U sin wt,U E m m >=为常数，求电感线圈上电流I 随时间的变化规律，设t=0时，I=0.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dtdI L . 由基尔霍夫定律知，闭合回路电压降为零. 于是 0dtdIL I R E =--. 改写为sin wt U L1L I R dt dIm +-=, 这是一个一阶线性非齐次方程. 记wt sin L Uq(t) ,L R p(t)m =-=, 由常数变易公式得到，)C ~dt sin wt LU e (e )C ~q(t)dt e(eI(t)m L RtLRt p(t)dtp(t)dt⎰⎰+=+⎰⎰=--.b a bt cos 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 22at a 22ib)t(a ib)t (a ibt at at +-=+⋅+-=+===++⎰⎰⎰22t LR m LRtm m LRt w(R/L) wt)cos w sin wt L R(e LU dt sin wt e LUdt sin wt L U e+-==⎰⎰令2222w(R/L)w φsin ,w(R/L)R/L φ cos +-=+=，于是由B sin A cos B cos A sin B)sin(A +=+知，22t LR mm LRt w (R/L)φ)sin(wt e L U dt sin wt L U e++=⎰，于是LRt 22m e C ~w(R/L)φ)sin(wt L U I(t)-+++=.再注意到初始条件I(0)=0，22m0022m w(R/L)φsin L U C ~0,C ~e e w (R/L)φsin LU I(0)+-==++=，因此，t LR22m22m ew(R/L)sin(φLUw (R/L)φ)sin(wt L U I(t)-+-++=).练习23. (1) 求dt bt cos e at ⎰； (2) 改写 t cos b sin t a +为θ)sin(t ba 122++，给出θ所满足的条件. (3) 由 Euler 公式b sin i b cos e ib+=和R b a, ,e e e b)i(a bi ai ∈=⋅+推导出:b asin sin b cos a cos b)cos(a b,sin a cos b cos a sin b)sin(a -=++=+和b))sin(a b)(sin(a 21b cos a sin -++=， b))cos(a b)(cos(a 21b cos a cos -++=.作业24. （1） 如例28中RC 电路图，设E=10V , R=100Ω, C=0.01 F, 开始时刻电容C 上电压为零并在此刻合上开关1，问经过多长时间电容C 两端电压为V 5U 1=?（2）如下RL 电路图，设E, R, L 均为正的常数，求开关闭合后电路中电流强度I(t)，假定I(0)=0.例30. 溶液混合问题：设容积为V （单位3m ）的密封容器装着某种溶液如下图，从A 以速度r （单位/s m 3）流入浓度为0C e >（常数）的相同溶液，经充分混合后在B 以相同速度r 流出容器, 假设时刻t=0时，容器溶液浓度为0，问容器中浓度随时间变化的规律.解：设时刻t 时容器溶液浓度为C(t)，且C(0)=0，则由溶质出入平衡，也即流入等于流出，由微元法建立如下等式：V C(t))Δt)(C(t C(t)Δt r C Δt r e -+≈-，即e C VrC V r dt dC +-=. (以下略)作业25. 假设伊利湖的存水量为34m 1048⨯，从休伦湖流入和从安大略湖流出的速度都是每年34m 1035⨯，在t=0时刻，伊利湖的污染物浓度时休伦湖的5倍. 如果流出的水是完全混合好的湖水，问使得伊利湖的污染物浓度减少到休伦湖2倍需要多少时间？（假定休伦湖污染物浓度为常数0C e >） 3. Bernoulli 方程及其解法称形如R n ,y q(x)y p(x)dxdyn ∈+=为Bernoulli 方程. 解法：当0y ≠时，改写原方程1n , n)q(x)(1y p(x) n)(1dxdy y n)-(1n -1n -≠-+-=, 令n)q(x)(1n)p(x)u (1dx du ,y u n 1-+-==-，这是一个一阶线性非齐次方程.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0y ≠时，改写原方程为 x 2)(1y x62)(1dx dy 2)y-(1212---=--，令21y u -=，则 x u x6dx du +-=. 由常数变易公式得到， 6276-dx x6dx x6xC8x C)dx x (x )C x dx e(eu(x )+=+=+⎰⎰=⎰⎰-.返回原变量得到62xC8x y 1+=.(2) 当y=0时，容易验证0y =也是原方程的解. 作业26. 求解方程（1）33y x y x dxdy=+； （2）1y(1) ,y x y '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）2y 2x y dx dy -=； （2）33y x x y 1dx dy -=. 解：(1) 这是一个一阶方程，非线性方程，不是Bernoulli 方程.(a) 当0y ≠时，交换自变量和因变量而改写原方程为 y x y2y y 2x dy dx 2-=-=. 这是一个一阶线性方程. 由常数变易公式得到， C)y)dy (e(ex dy y2dy y2+-⎰⎰=⎰-，即 |)y |ln (C y C)y)dy (y1(y x 222-=+-=⎰为所求方程的通积分. (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）3yx ydx dy +=； （2） y 2y x dx dy 22+=. 5. 一些一阶线性方程的理论 （1）考虑方程q(x)y p(x)dxdy=+，其中p(x), q(x)都是以w>0为周期的连续函数. 用常数变易公式证明：（a) 若0q(x)≡，则方程任一非零解都以w 为周期的周期函数充要条件是p(x)的平均值.0p(x)dx w1(x)p w0==⎰ (b) 若q(x)不恒为零，则方程有唯一w 周期解充要条件是0p(x)dx w1(x)p w0≠=⎰, 试求出此解. （参见丁同仁、李承治《常微分方程教程》P36 习题5， 6）。
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dy 2 y x
dx
2
dx x 2
除此之外，上述方法还具有更强的简易性与广泛性。当方程右端为多个
式子组成的更复杂的函数时，我们仍然可以使用此方法得到通解。
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例题2： dy(x1)2(4y1)28xy1
暨南
dx
第开
四大 届学 大第 学八
dy ( x 1)2 dx
y
1(x 3
1)3
c1
生届 “大 数学 学生 之文
n阶常系 数非齐次 线性方程
解法： d dtnn xa1d dtn n1 x 1… an 1d dx tanx0
d d t n n x a 1d d t n n 1 x 1 ... a n 1d d x t a n xf1t
d d t n n x a 1d d t n n 1 x 1 … a n 1d d x t a n xf2 (t)
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d d t n n x a 1 d d t n n 1 x 1 … a n 1 d d x t a n x f1 ( t) f2 ( t) … fm ( t)
d d n n xta 1d d n n 1 1 x t a n 1d d x ta nxf1(t) d d n n xt a 1d d n n 1 1 x t a n 1d d x ta nxf2(t)
x (t)
+
x 1 t 通解
+
x2 t
这种形式的方程与我们所研究的一阶线性非齐次常微分方程
dy pxyqx
dx
有某种程度的相似性
我们是否可以按上 述方法将其分解为 两个方程分别求解 ，最终得到通解呢
？
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暨南 第开 四大 届学 大第 学八 生届 “大 数学 学生 之文 美化 ”素 论质 坛教
形式不同的结果之间只相差一个常数。
• 着力去挖掘隐藏在结论背后的理论与方法
间 本质的联系 • 数学的魅力在于探求结果过程中 逻辑思
维 的运用，联想能力 的拓展与 锲而不舍 的钻研精神 三者的融会贯通
• 学会的不仅是一种方法，而是一种 思想
• 数学之中体现 哲学真理
• 不断 尝试，不断 求索，就会加深理解，
步骤一： 设 y ux
之文 美化 ”素 论质
步骤二：
dy u x du
dx
dx
此时，u已被视为x的函数
坛教 育 节
dy
将 d x 代回原式，就得到关于u与x的方程，从而求解
u
y
x
c
ye pxdx
y ce pxdx
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暨南 第开
d yd cxep xd x cxpxep xd x
d x d x
暨南 第开 四大 届学 大第 学八 生届 “大 数学 学生 之文 美化 ”素 论质 坛教
育 节
浅析常数变易法
南开大学 经济学院 06级金融学系
张婷
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暨南
第开
四大 届学
常数变易法是求解一阶非齐次线性常微分方程的重要方法，即将常数
大第 学八
变易为待定函数，通过求解待定函数的表达式进而求出原方程通解
育 节
经过推导，验证了我们的猜想。具体过程如下：
dy p x y
dx
dy q x
dx
y1 x cepxdx + y2xqxdx
ycepxdx qxdx
yepxd x(cqxe pxd xd x) 变形： y c e p x d x e p x d x qxe p x d x d x
暨南
第开 四大 届学 大第
例题1： dy 2 y x dx x 2
学八 生届 “大 数学
dy 2 y dx x
y ec1 x 2 (c1为任意常数）
学生
+
之文 美化 ”素 论质
dy x dx 2
y
1 4
x2
c2
(c2为任意常数)
坛教 育
将上述结果进行求导：
节
dy 2ec1 x x
e c1 y x2
拓宽思维的深度与广度
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暨南 第开 四大 届学 大第 学八 生届 “大 数学 学生 之文 美化 ”素 论质 坛教
dy u x du
dx
dx
四大
届学
大第
学八 生届
此时，二者的本质相同，只是常数变易法中的变量代换更为复杂，不易辨别。
“大
常数变易法的目的是将原方程变换为只含c x 与 x 的方程。从而求出 c x
数学
学生
之文 美化 ”素
y cxepxdx
yep xd x(cqxe p xd xd x)
yc ep xd x1
px
qxe p xd xd ep xd x
ycepxdxp1x qxd lnepxdx
ycepxdxp1xqxdpxdx
ycepxdx qxdx
可见，通过此方法求得的结果与常数变易法结果实质上是一致
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的。我们也可以通过实际例题来加以验证，结论亦成立。
论质
坛教
育
节
常数变易与变量代换是相互渗透相互联系的 齐次的或可化为齐次的方程中所代换的是一个变量
而一阶线性非齐次常微分方程中，由于方程的复杂性， 因此被代换的是一个表达式
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暨南 第开 四大 届学 大第 学八 生届 “大 数学 学生 之文 美化 ”素 论质 坛教
育 节
常数变易法与迭加原理
d d tn n x a 1d d tn n 1 x 1 … a n 1d d x t a n xf1 (t) f2 (t)
dy (4 y 1)2 dx
11
y
16(xc2) 4
美化 ”素 论质
dy 8xy dx
y e4x2 c3
坛教 育 节
dy 1 dxΒιβλιοθήκη y x c4四式代数和即为原方程通解！此方法大 大简化了运算过程，从而降低了运算量 ！
……
……
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暨南 第开 四大 届学 大第 学八 生届 “大 数学 学生 之文 美化 ”素 论质 坛教
x1 t
x2 t
d d nn xta 1d d n n 1 1 x t a n 1d d x ta nxfm (t)
xn t
各方程特解与齐次方程通解的代数和即为原n阶非齐次方程的通解
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暨南 第开 四大 届学 大第 学八 生届 “大 数学 学生 之文 美化 ”素 论质 坛教
育 节
由此，对于最初所研究的一阶线性常微分方程，又增 加了一种十分重要的解法。虽然结果形式上不同，但本质 相同。
生届
“大
数学 学生 之文
例如：
dy pxyqx
美化
dx
”素
论质 坛教
育 节
解法： dy p x y
dx
y ce pxdx
y cxepxdx
cx
yepxdx(cqxepxdxdx)
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暨南 第开
常数变易法的本质
四大
届学 大第 学八 生届
解法：
齐次方程：
dy dx
g
y x
“大
数学 学生