第一章_一阶微分方程的应用

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0.05mg/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,Model3:Populationdynamics,Inthissectionweexamineequationsoftheformy=f(y),calledautonomousequations,wheretheindependentvariabletdoesnotappearexplicitly.,Themainpurposeofthissectionistolearnhowgeometricmethodscanbeusedtoobtainqualitativeinformationdirectlyfromdifferentialequationwithoutsolvingit.,Simplestmodel:populationgrowthrateisproportionaltocurrentsizeofthepopulation:,Solution:exponentialgrowth):,Model3:PopulationdynamicsLogisticGrowth,Anexponentialmodely=ry,withsolutiony=ert,predictsunlimitedgrowth,withrater0independentofpopulation.,Assuminginsteadthatgrowthratedependsonpopulationsize,replacerbyafunctionh(y)toobtainy=h(y)y.,Wewanttochoosegrowthrateh(y)sothath(y)rwhenyissmall,h(y)decreasesasygrowslarger,andh(y)0.,Ourdifferentialequationthenbecomes,ThisequationisknownastheVerhulst,orlogistic,equation.,Thelogisticequationfromthepreviousslideis,Thisequationisoftenrewrittenintheequivalentform,whereK=r/a.Theconstantriscalledtheintrinsicgrowthrate,andaswewillsee,Krepresentsthecarryingcapacityofthepopulation.,Adirectionfieldforthelogisticequationwithr=1andK=10isgivenhere.,Equilibriumsolutionsofthelogisticequation,Ourlogisticequationis,Twoequilibriumsolutionsareclearlypresent:,Indirectionfieldbelow,withr=1,K=10,notebehaviorofsolutionsnearequilibriumsolutions:y=0isunstable,y=K=10isasymptoticallystable.,Qualitativeanalysisofthelogisticequation,Tobetterunderstandthenatureofsolutionstoautonomousequationsy=f(y),westartbygraphingf(y)vs.y.,Inthecaseoflogisticgrowth,thatmeansgraphingthefollowingfunctionandanalyzingitsgraphusingcalculus.,Qualitativeanalysis,criticalpoints,Theinterceptsoffoccuraty=0andy=K,correspondingtothecriticalpointsoflogisticequation.,Thevertexoftheparabolais(K/2,rK/4),asshownbelow.,Qualitativeanalysis,increasing/decreasing,Notedy/dt0for0yK,soyisanincreasingfunctionoftthere(indicatewithrightarrowsalongy-axison0K).,Inthiscontextthey-axisisoftencalledthephaseline.,Qualitativeanalysis,concavity,Next,toexamineconcavityofy(t),wefindy:,Thusthegraphofyisconcaveupwhenfandfhavesamesign,whichoccurswhen0K.,Thegraphofyisconcavedownwhenfandfhaveoppositesigns,whichoccurswhenK/2yK.,Inflectionpointoccursatintersectionofyandliney=K/2.,Qualitativeanalysis,curvesketching,Combiningtheinformationonthepreviousslides,wehave:,Graphofyincreasingwhen0K.Slopeofyapproximatelyzerowheny0oryK.Graphofyconcaveupwhen0K.GraphofyconcavedownwhenK/2yK.Inflectionpointwheny=K/2.,Usingthisinformation,wecansketchsolutioncurvesyfordifferentinitialconditions.,Qualitativeanalysis,curvesketching,Usingonlytheinformationpresentinthedifferentialequationandwithoutsolvingit,weobtainedqualitativeinformationaboutthesolutiony.,Forexample,weknowwherethegraphofyisthesteepest,andhencewhereychangesmostrapidly.Also,ytendsasymptoticallytotheliney=K,forlarget.,ThevalueofKisknownasthecarryingcapacity,orsaturationlevel,forthespecies.,Notehowsolutionbehaviordiffersfromthatofexponentialequation,andthusthedecisiveeffectofnonlinearterminlogisticequation.,Model3:PopulationdynamicsExactsolution:separatingvariables,Providedy0andyK,wecanrewritethelogisticODE:,Expandingtheleftsideusingpartialfractions,Thusthelogisticequationcanberewrittenas,Integratingtheaboveresult,weobtain,Exactsolution:resolvingforexplicitsolution,Wehave:,If0y0K,then0yKandhence,Rewriting,usingpropertiesoflogs:,Exactsolution:resolvingforexplicitsolution,Wehave:,for0y0K.Also,thissolutioncontainsequilibriumsolutionsy=0andy=K.,Hencesolutiontologisticequationis,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，求在任意时刻的水面高度和将水放空所需的时间,问题分析,从