《阶微分方程的应用》PPT课件.ppt

1.8 一阶微分方程的应用 应用微分方程去解决一些实际问题 应用介绍： 应用大意 应用一： 曲线族的等角轨线 应用二： 雨滴的下落 应用三： 人口增长模型 应用四： 静脉注射给药 应用五： 水流问题,1： 应用大意,适应范围 与变化率有关的各种实际问题 应用三步曲 (1) 建立模型 Modelling (2) 模型求解 Solving (3) 模型应用 Application 建议: 模型要详略得当,应用一：曲线族的等角轨线,设给定一个平面上以C为参数的曲线族,（*）,我们设法求出另一个以k为参数的曲线族,（*）,使得曲线族（*）中的任一条曲线与曲线族,样的曲线族（*）是已知曲线族（*）的,等角轨线族。,正交轨线族。,的正交轨线族。,设 y=y(x) 为(C) 中任一条曲线，于是存在相应的C，使得 因为要求x,y,y 的关系，将上式对x求导数，得 (1.84) 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,解：对方程两边关于x求导得,正交，故满足方程,曲线族为,这是一个椭圆，如右图,放大此图,图2.16,应用二： 雨滴的下落,考虑雨滴在高空形成后下落的过程中速度的变化 三种不同的假设 (1) 自由落体运动 (2) 小阻力的情况 (3) 大阻力的情况,(1)自由落体运动,下落过程中没有任何阻力,小阻力的情况,下落过程中阻力与速度和半径的乘积成比例,(3)大阻力的情况,下落过程中阻力与速度和半径的乘积平方成比例,三、药物设计,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。 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。,一次给药的药时曲线,治疗窗口,药物消除类型,1 一级动力学消除（恒比消除）： 单位时间内按血药浓度的恒比进行消除。消除速度与血药浓度成正比。 若以血药浓度（C）的对数与时间（t）作图，为一直线。,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行消除。消除速度与血药浓度无关。,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（混合型消除）：,模型及其数值实现,阅读材料: 服药问题,医生给病人开处方时必须注明两点:服药的剂量和服药的时间间隔.超剂量的药品会对身体产生严重不良后果,甚至死亡,而剂量不足,则不能达到治病的目的.已知患者服药后,随时间推移,药品在体内逐渐被吸收,发生生化反应,也就是体内药品的浓度逐渐降低.药品浓度降低的速率与体内当时药品的浓度成正比.当服药量为A、服药间隔为T,试分析体内药的浓度随时间的变化规律.,体内药的浓度随时间的变化规律,Model 3: Population dynamics,In this section we examine equations of the form y = f (y), called autonomous equations, where the independent variable t does not appear explicitly.,The main purpose of this section is to learn how geometric methods can be used to obtain qualitative information directly from differential equation without solving it.,Simplest model: population growth rate is proportional to current size of the population:,Solution: exponential growth):,Model 3: Population dynamics Logistic Growth,An exponential model y = ry, with solution y = ert, predicts unlimited growth, with rate r 0 independent of population.,Assuming instead that growth rate depends on population size, 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 y grows larger, and h(y) 0 when y is sufficiently large.,The simplest such function is h(y) = r ay, where a 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 rate, and as we will see, K represents the carrying capacity of the population.,A direction field for the logistic equation with r = 1 and K = 10 is given here.,Equilibrium solutions of the logistic equation,Our logistic equation is,Two equilibrium solutions are clearly present:,In direction field below, with r = 1, K = 10, note behavior of solutions 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 autonomous equations y= f(y), we start by graphing f (y) vs. y.,In the case of logistic growth, that means graphing the following function and analyzing its graph using calculus.,Qualitative analysis, critical points,The intercepts of f occur at y = 0 and y = K, corresponding to the critical points of logistic equation.,The vertex of the parabola is (K/2, rK/4), as shown below.,Qualitative analysis, increasing/decreasing,Note dy/dt 0 for 0 y K, so y is an increasing function of t there (indicate with right arrows along y-axis on 0 y K).,Similarly, y is a decreasing function of t for y K (indicate with 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 examine concavity of y(t), we find y:,Thus the graph of y is concave up when f and f have same sign, which occurs when 0 K.,The graph of y is concave down when f and f have opposite signs, which occurs when K/2 y K.,Inflection point occurs at intersection of y and line y = K/2.,Qualitative analysis, curve sketching,Combining the information on the previous slides, we have:,Graph of y increasing when 0 K. Slope of y approximately zero when y 0 or y K. Graph of y concave up when 0 K. Graph of y concave down when K/2 y K. Inflection point when y = K/2.,Using this information, we can sketch solution curves y for different initial conditions.,Qualitative analysis, curve sketching,Using only the information present in the differential equation and without solving it, we obtained qualitative information about the solution y.,For example, we know where the graph of y is the steepest, and hence where y changes most rapidly. Also, y tends asymptotically to the line y = K, for large t.,The value of K is known as the carrying capacity, or saturation level, for the species.,Note how solution behavior differs from that of exponential equation, and thus the decisive effect of nonlinear term in logistic 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 obtain,Exact solution: resolving for explicit solution,We have:,If 0 y0 K, then 0 y K and hence,Rewriting, using properties of logs:,Exact solution: resolving for explicit solution,We have:,for 0 y0 K.,It can be shown that solution is also valid for 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数据吻合,水的流出问题,一横截面积为 A，高为 H 的水池内盛满了水，有池底一横截面积为 B 的小孔放水。设水从小孔流出的速度为 ，