第一章一阶微分方程的应用

1、应用一：曲线族的等角轨线应用一：曲线族的等角轨线设给定一个平面上以设给定一个平面上以c c为参数的曲线族为参数的曲线族( , ,)0f x y c （* *） 我们设法求出另一个以我们设法求出另一个以k k为参数的曲线族为参数的曲线族( , , )0g x y k （* * *） 使得曲线族（使得曲线族（* * *）中的任一条曲线与曲线族）中的任一条曲线与曲线族样的曲线族（样的曲线族（* * *）是已知曲线族（）是已知曲线族（* *）的）的（* *）中的每一条曲线相交时成定角）中的每一条曲线相交时成定角 则称这则称这等角轨线族。等角轨线族。当当 2时，称曲线族（时，称曲线族（* * *）是（）

2、是（* *）的）的正交轨线族。正交轨线族。例如：曲线族例如：曲线族 0ykx是曲线族是曲线族 2220 xyc的正交轨线族。的正交轨线族。 ( )y y x1( )yy xa1a2a设设 y=y(x) 为为(c) 中任一条曲线，于是存在相应中任一条曲线，于是存在相应的的c，使得，使得 因为要求因为要求x,y,y 的关系，将上式对的关系，将上式对x求导数，得求导数，得 (1.84)这样，将上两式联立，即由这样，将上两式联立，即由 上述关系式成为曲线族满足的微分方程上述关系式成为曲线族满足的微分方程例例1 1 求抛物线族求抛物线族 2ycx的正交轨线族。的正交轨线族。解：解：对方程两边关于对方程两

3、边关于x x求导得求导得2dycxdx由由 2ycx解出解出c c代入上式得曲线族代入上式得曲线族 2ycx在点在点 ( , )x y处切线斜率为处切线斜率为 2dyydxx由于所求曲线族的曲线与由于所求曲线族的曲线与 2ycx中的曲线在中的曲线在 ( , )x y正交，故满足方程正交，故满足方程 2dyxdxy 这是一个变量可分离方程求解得这是一个变量可分离方程求解得 2ycx的正交的正交曲线族为曲线族为2222xyk这是一个椭圆，如右图这是一个椭圆，如右图xy放大此图 图图2.162.16xy22,(0)0,( )101( )52500031.6,316/1137.6/dvmmgvdtv

4、tgtts tgttsmtvm skm h时，2122/0.081,(0)0,(0)0,( )10.001121 1lim ( )121/435.6/kt rttdvmmgk rvvdtdvkgvvdtrr gv terkev tm skm h（取）（取定常数）2222/,(0)0,(0)0,1( )10.001,lim ( )7/25.2/grtgrttdvmmgk r vvdtdvgvvdtrgrev terv tm skm h取取定常数后医生给病人开处方是必须注意两点：医生给病人开处方是必须注意两点： 服药的剂量和服药的时间间隔。服药的剂量和服药的时间间隔。 超剂量的药物会对患者产生严重

5、不良后果，超剂量的药物会对患者产生严重不良后果， 甚至死亡；剂量不足，则不能达到治疗的效果。甚至死亡；剂量不足，则不能达到治疗的效果。 一次给药的药时曲线一次给药的药时曲线血药浓度血药浓度mg/l时间时间残留期残留期持续期持续期药峰时间药峰时间潜伏期潜伏期药峰浓度药峰浓度最低中毒浓度最低中毒浓度最低有效浓度最低有效浓度安全范围安全范围转化排泄过程转化排泄过程多次给药的药时曲线多次给药的药时曲线0 01 11 12 22 23 34 45 56 6血血药药浓浓度度时间时间cmaxcmaxcmincmin治疗窗口治疗窗口1 一级动力学消除（恒比消除）：一级动力学消除（恒比消除）： 单位时间内按血药

6、浓度的恒比进行消单位时间内按血药浓度的恒比进行消除。除。消除速度与血药浓度成正比。消除速度与血药浓度成正比。 若以血药浓度（若以血药浓度（c）的对数与时间）的对数与时间（t）作图，）作图，为一直线。为一直线。0ek tetdck ccc edt 2. 零级动力学消除（恒量消除）：零级动力学消除（恒量消除）： 单位时间内始终以一个恒定的数量进单位时间内始终以一个恒定的数量进行消除。行消除。消除速度与血药浓度无关。消除速度与血药浓度无关。0tdckccktdt 是指包括零级和一级动力学消除在内的混合型消除方式。如当药物剂量急剧增加或患者有某些疾病，血浓达饱和时，消除方式则可从一级动力学消除转变为零

7、级动力学消除。如乙醇血浓0.05 mg/ml时，则可转成按零级动力学消除。 3米氏消除动力学（米氏消除动力学（混合型消除混合型消除）：模型及其数值实现模型及其数值实现model 3: population dynamicsin this section we examine equations of the form y = f (y),called autonomous equations, where the independentvariable t does not appear explicitly.the main purpose of this section is to lea

8、rn how geometricmethods can be used to obtain qualitative informationdirectly from differential equation without solving it.simplest model: population growth rate is proportional tocurrent size of the population:solution: exponential growth):model 3: population dynamicslogistic growth an exponential

9、 model y = ry, with solution y = ert, predictsunlimited growth, with rate r 0 independent of population. assuming instead that growth rate depends on populationsize, replace r by a function h(y) to obtain y = h(y)y. we want to choose growth rate h(y) so that h(y) r when y is small, h(y) decreases as

10、 y grows larger, and h(y) 0. our differential equation then becomes this equation is known as the verhulst, or logistic, equation. the logistic equation from the previous slide is this equation is often rewritten in the equivalent form where k = r/a. the constant r is called the intrinsic growth rat

11、e, and as we will see, k represents the carrying capacity of the population. a direction field for the logisticequation with r = 1 and k = 10is given here.equilibrium solutions of the logistic equation our logistic equation is two equilibrium solutions are clearly present: in direction field below,

12、with r = 1, k = 10, note behavior ofsolutions near equilibrium solutions:y = 0 is unstable,y = k=10 is asymptotically stable.qualitative analysis of the logistic equation to better understand the nature of solutions to autonomousequations y= f(y), we start by graphing f (y) vs. y. in the case of log

13、istic growth, that means graphing thefollowing function and analyzing its graph using calculus.qualitative analysis, critical points the intercepts of f occur at y = 0 and y = k, correspondingto the critical points of logistic equation. the vertex of the parabola is (k/2, rk/4), as shown below.quali

14、tative analysis, increasing/decreasing note dy/dt 0 for 0 y k, so y is an increasing function oft there (indicate with right arrows along y-axis on 0 y k (indicatewith left arrows along y-axis on y k). in this context the y-axis is often called the phase line.qualitative analysis, concavity next, to

15、 examine concavity of y(t), we find y: thus the graph of y is concave up when f and f have samesign, which occurs when 0 y k. the graph of y is concave down when f and f have oppositesigns, which occurs when k/2 y k. inflection point occurs at intersection of y and line y = k/2.qualitative analysis,

16、 curve sketching combining the information on the previous slides, we have: graph of y increasing when 0 y k. slope of y approximately zero when y 0 or y k. graph of y concave up when 0 y k. graph of y concave down when k/2 y k. inflection point when y = k/2. using this information, we cansketch sol

17、ution curves y fordifferent initial conditions.qualitative analysis, curve sketching using only the information present in the differential equationand without solving it, we obtained qualitative informationabout the solution y. for example, we know where the graph of y is the steepest,and hence whe

18、re y changes most rapidly. also, y tendsasymptotically to the line y = k, for large t. the value of k is known as the carrying capacity, orsaturation level, for the species. note how solution behavior differsfrom that of exponential equation,and thus the decisive effect ofnonlinear term in logistic

19、equation.model 3: population dynamics exact solution: separating variables provided y 0 and y k, we can rewrite the logistic ode: expanding the left side using partial fractions, thus the logistic equation can be rewritten as integrating the above result, we obtainexact solution: resolving for expli

20、cit solutionwe have:if 0 y0 k, then 0 y k and hencerewriting, using properties of logs:exact solution: resolving for explicit solutionwe have:for 0 y0 k. also,this solution contains equilibrium solutions y = 0 and y = k. hence solution to logistic equation is模型预测的动态行为与大量的实验和观测模型预测的动态行为与大量的实验和观测数据吻合数据吻合 水的流出问题水的流出问题一横截面积为一横