《微积分》课程教学课件（Calculus）09. First-Order Differential Equations

CALCULUSTHOMAS'CALCULUSARUYTRANSCENChapter 9JAMESSTEWARTFirst-Order DifferentialEquationsERMNNtCIHeAA9.11,GeneralFirst-OrderDifferential Equations andSolutions, SlopeFields,andSolutionsEuler'sMethod国嘉EXAMPLE1Show that every member of the family of functionsAfirst-order differentialequatiomisanequationy=f+2案-stial equatiotin which f(x,y) is a function of two variables defined on a region in the xy-plane. The黑-12-g)equation is of firsr order because it imvolves only the first derivative dy/dr (and notoe the interval (0, oe), μhere C is any constanthigher-orderderivatives).WepointoutthattheequationsSelutonDiflerentiating y =C/x + 2gives=andy= f(x,y)2-c%（)+0--5are equivalent to Equation (1)We need to show that the difflerolion is saitisfied whien we substitute imo it the expressions (C/) +2for y,and C/rfor dly/dr.That ik, we need to verity than for allFe(0,00),The general solution to a frst-order differential equation is a-S-1-($+2)solution that contains all possiblesolutionsThis last sht-hand sidefoloexn1/2- (+2)]-(-)--号白Therefore, for every'value of C, the function y C/x + 2 is a solution of the differentia00

2016/11/15 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 9 First-Order Differential Equations Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.1 Solutions, Slope Fields, and Euler’s Method Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley 1. General First-Order Differential Equations and Solutions The general solution to a first-order differential equation is a solution that contains all possible solutions

EXAMPLE2Show that the function2F =6+1b(aa= (x + 1) -e-The solution satisfying the initial condition y(xo)-y is theisasto the fiinitial valuae probilem1particukar solution,d(0)=-A first-order iniial value problkem is a differential equation y'=f(x,y)SolutionThe equationwhose solution must satisfy an initial condition y(xo)=ye -MURE 4.1 Graph of th岁一岁一*aitialisfioerfialutiith-是(+1-)-1-1Om thelgfsideaftheequationOn the righr sideof the egs-x=(+)-1e-x=1-1ebefinctionsatisfies the initialconditionbecause国0) -[a+ 1) --1 -1----3,9.21.The Linear Equation'sStandard FormFirst-Order Linear EquationsSlide 4-9Slide 4- 10A first-order linear differential equation is one that can be written in the form+ Py-00.(3)where P and Q are continoous functions of x, Equation (I) is the linear equation's standard form.EXAMPLE1Put the following equation in standard form:2.Solving Linear Equationsdy-2+3x>0Solution*2-2+3会tivide by's会--Stanitandfonnwith'a)=3/aadSlide 4-11Slide 4.12

2016/11/15 2 The solution satisfying the initial condition y(x0 )=y0 is the particular solution. A first-order initial value problem is a differential equation y’=f(x,y) whose solution must satisfy an initial condition y(x0 )=y0 . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 9 9.2 First-Order Linear Equations Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 10 1. The Linear Equation’s Standard Form Slide 4 - 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 12 2. Solving Linear Equations

We solve the equation2(0) -+ Py9+Pa-0a)#+→毫+Pby multiplying both sides by a positive fiunction s(x) that trainsforms the lef-hand side intotstratnRilaxthe derivative oftheproduct w(x) -y.Wewillshowhowto find v ina momcmt, but first weu-PyThetomeC*+ Pby 0tl)Thislast equation will boldi+P()g()MiolybyasitheetahPtt wchoin semieL[(0)]-0(x)g(s)鲁-Pdtalinm=>0treegrate wite reopedi(x).y(x)0(x) ds1[鲁-/rdESx)dr(2)Vu(x)law-[pd90Cahiesigra in lsntegratingCactorchmefrsEspchethideriuaise Sr!w=e/relide 4-13Slide 4- 14Next we muliply both sides of the stindand form by a(x) and integrate:To. solve the linear equation y' + P(x)y = O(x), multiply both sides by the (离-)-*itegating faco() eand inegrate both sides.--#EXAMPLE2Solve the equation宗一+3Ldhy(5)-x>0.SolutionFirst we pat the equation in standard form (Example 1):-/Inge teath sida一Jy=-1+c.so P() 3/x is identified.Solving this lastquatiofogiesthgeealsolutioThe integrating faclor isu(x) -e/ rd= efi-/ndJ=r(-++c)--++c),x > 0Cattofiebaations0E:eso ws as simple aposiblc=ear1>0er-:Slide 4- 15Slide 4- 18EXAMPLE3Find the particular solution of3x/ y = tsx +.1,x > 0,ExercisesFirst-Order Linear EquationsSothdffialqutionmErcissatisfying (1) = 22.0岁+2ey-1ty-hrt!1+1>0With x > 0, we write the equation in standar ySolutienThenthe integntingfactorisgheef-apam-sinx3. g + 3y ->1>0→ny-ungndx-ny-/(nr+r-ThusSolving Initial Value ProblemsIntegration by parts of the righe-hand side x-1/ly+C=-x(inx +1)Solve the imitial value problems in Exercises 1520Thereforor-ly - --/(nr + 1) 3r-V9 + C15.岁+2g-3, 3K0)-1, solving for y = (Inx + 4) + Cr/p,When x = 1 and y. = 2 this last cequntion bec2 = (0 + 4) + C,16岁+20 2)-180C-2Substitution into the eqiation for y gives the particular solhution17. 需+o,0>0,(/2)-1y = 2r/ Inx = 4.Slide 4- 17Slide 4-18

2016/11/15 3 Slide 4 - 13 integrating factor Slide 4 - 14 Slide 4 - 15 Slide 4 - 16 Slide 4 - 17 Slide 4 - 18 Exercises