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# 548 Class Note for MA 26600 with Professor Yu at Purdue

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

A Brief Review Chapter 1 Introduction Chapter 2 First Order Differential Equations 1 Chapter 1 Introduction There is nothing solid in Chapter 1 However sometimes when you can7t gure out how to solve an equa tion or system with methods learned in other chapters you may try 1 Classi cation a first order linear the most important one integrating factor or formula in Sec21 b first order nonlinear the only nonlinear one a separable homogenous or exact c high order linear methods in Chapter 34 or Chapter 6 Laplace transform d linear system a Chapter 7 2 sketch the direction field of a given equation see which answer solution fits the direction field 3 directly verify a solution by plugging it in the equation 2 Chapter 2 First Order Differential Equations 1 split the content of this chapter into three catolog see the subsection below Note that Section 27 Numerical Approximation Euler7s Method is not required Remember the formulae 1 list below at least do two practice problems for each topic or go over corresponding homework If you can solve a certain differential equation also depend on if you are familar with the basic inte grals You can find some review material on integrals on my course webpage the bottom of Update List 21 How to solve a rst order equation 0 First Order Linear Equation Section 21 plttgtyglttgt lt1 Step 1 Calculate the integrating factor mt exp plttgtdt lt2 m l lttgtglttgtdtcl lt3 Step 2 Calculate the solution Eqns 2 and 3 give an explicit formula of the solution to equation 1 which is the MOST IMPORTANT one in this course Probably you will use it again in other math or physics courses Practice Problems Section 21 1 15 16 o Separable Equation Section 22 dz Cy Mo Ny 0 or Mo dz Ny dy 0 or The equation can be solved by first separating terms and then take the direct integration SECTION 2 Practice Problems Section 22 1 14 Homogeneous Equation Section 22 d digfr7yyyI 4 Step 1 Introduce a new depenent variable 1 so that v yz or yz 1 which lead to dy7dzv7 d1 3 dx Ia lt5 Step 2 In equation 4 replace yx by v by v x37 we get dv U IE 91 or dv IE 91 v 6 This is a separable equation can be solved by moving all the 1 terms to the left side and all the 1 terms to the rightnote that you can7t put the dv or d1 terms in denominator and then taking the integration ie dv dz 7 7 5 91 7 v 1 Practice Problems Section 22 32 36 Exact Equation See Example 2 of section 26 if you don7t like the general formulation A first order equation MI7yNI7yy 0 7 is an exact equation if and only if Myr7yNxI7y 8 Procedure to solve an equation of form 7 Step 1 check if 7 is an exact equation by verify 8 if true then there is a functino z y such that ML y MW 9 lt1gtyr7yNr7y Step 2 Take integration of Mz y with respect to I holding y constant we get gtz7yQz7yhy7 whereQz7yMz7ydr lt9 Step 3 Take partial derivative of z y with respect to y and let it equal to Nx y ltIgtyx7 y am y h y NW y then solving for h y we have h yNI7yr Qy17y 10 Step 4 The right side of equatin 10 should be a function of y only taking the integration on the both side we get an expression of hy plugin the value of hy in equation 9 and eventu ally an implicit solution is given by ay QI7yhy 0 Practice Problems Section 26 3 7 13 CHAPTER 2 FIRST ORDER DIFFERENTIAL EQUATIONS 3 22 The Existence and Uniqueness Theorem 1 Knowing how to determine the interval in which the solution is certain to exist Section 24 Note that you must convert the equation in following form the coeP cent of y term is 1 y oty y Practice Problems Section 24 1 3 2 That7s all 2 3 Applications 1 Salt water mixing in a tank know how to build a governing equation Section 2 example 1 d 7Q rate in 7 rate out dt Practice Problems Section 23 2 3 2 Population dynamicsSecti0n25 How to nd the equilibrium solutions of an autonomous equa tionzeros of fy Determine the stability of the equilibrium solutions by drawing some arrows on phase line There are several kind of stability stable including asymptotically stable unstable including semistable see Problem 9 section 25 Practice Problems Section 25 3 9

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