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# Ordinary Differential Equations MA 26600

Purdue

GPA 3.97

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This 3 page Class Notes was uploaded by Dorothea Bode on Saturday September 19, 2015. The Class Notes belongs to MA 26600 at Purdue University taught by Haijun Yu in Fall. Since its upload, it has received 38 views. For similar materials see /class/208140/ma-26600-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15

A Brief Review Chapter 3 Second Order Linear Equations Chapter 4 Higher Order Linear Equations If you have time for each iterm I list below you should do at least two practice problems 1 Chapter 3 Second Order Linear Equations 1 Knowing how to determine the longest interval in which the solution to the initial value problem y pty WM 907 We yo Mia y exists without solving the initial value problem Answer the longest interval containing to in which pt qt and gt are continuous If the coef cients of y in the given equation is not one you need first divide the equation by that coef cients Practice Problems Section 32 8 9 3 Homogeneous Equations with constant coef cients ay by cy0 1 First solve the characteristic equation which is aT2bTc0i 2 Using quadratic formula the roots are given by 77bvb274ac T77b7vb274ac 7 2a 7 27 2a T1 There might be three cases i 7 1 T2 are real and different rootswhen 2 7 4ac gt 0 then the general solution of 1 is given by yt cl er1t 0262 Practice Problems section 31 14 10 7 17 2 are complex conjugateswhen 2 7 4 ac lt 0 suppose 7 1 A 21 T2 A 7 iM the two real valued solutions are which are real part and imaginary part of EO HME ut eMcosMt vt emsinm i The the general solution is given by yt cl 6 cosm 02 e sinm Practice Problems Section 34Section 33 of 9th edition 10 17 18 real repeated rootswhen 2 7 4ac 0 7 1 T2 7 i T in this case the general solution is given by yt cl en 02 ten Practice Problems Section 3534 in 9th edition 3 12 3 Nonhomogeneous Equations General form y pty 4tyyt 3 2 SECTION 2 With constant coef cients ay by cyyt7 CNN 4 The general solution of 3 or can be written as Mt 0191M 0292t Yt7 where Cly1t 52y2t is the general solution of the corresponding homogeneous equation and Yt is a particular solution of the nonhomogeneous equation There are two methods to find a partic ular solution Yt i Undetermined Coefficients IMPORTANT This method is applicable to equation 4 when gt is an exponential function sine cosine function polynomial function or their product or sum The key step is to choose the right form of a particular solutino Yt Table 361 on page 181 of 8th edition textbookls is Table 351 on 9th edition tell you how to choose the right form Please do go over all the homework of this section Variation of Parameters This method is more powerful because it is applicable to the general equation Suppose you can find two solutions y1t and y2t of the corresponding homogeneous equation with nonzero Wronskian Wy1 y2t LIl 0 then a particular solution Yt of the nonhomogeneous equation is given in following form YO u1tyit WW 92037 5 where 7 y2lttgtglttgt u y1lttgtglttgt W Wimmdt 2 lemmdt 6 Remember this formula and do at least one practice problem You wont have enough time to repeat the full procedure of variation of parameters in exam Practice Problems Section 37 36 in 9th edition 7 10 13 9 Application Spring Mass system the governing equation of a spring mass system mum W t wt Flttgt lt7 where m is the mass 7 is the damping coef cient and k is the spring constant Pay special atten tion to their units when you build the eqaution You should know how to find the amplitude period or quasi period of the solution and know the differences between 70 and 7 0 You needn7t remember any formula except the governing equation Practice Problems Section 38 37 in 9th edition 5 9 2 Chapter 4 Higher Order Linear Equations This chapter is an extension of Chapter 3 which means most of the results of second order linear equa tions can be directly extended to higher order linear equations Following are the differences 1 For a higher order linear equation Llyl aoy a1y 1an71y any0 8 its characteristic equation is aornalrn 1an1ran0 9 CHAPTER 4 HIGHER ORDER LINEAR EQUATIONS You need know how to factorize the polynomial on the left side of Some easy examples T33r23r1 013 r1r2r371 T373T23r7l T703 4T1T2T3l T42T2l 7 212 mr139r2i T3T47i factors ofan Generally you can try find a root by looking at m for example for T4T377T27T60 10 the factors of anwhich is 6 in this example are i 1 i 2 i 3 i 6 The factors of a0 which is 1 here are i1 So you can try 1 7 1 2 7 13 7 36 7 6 check which one is a root we found r 1 7 1 2 7 3 are the four roots Practice Problems Section 42 11 29 F0 repeated roots when n gt 2 in equation 8 the multiplicity of a repeated roots of characteristic equation could be larger than 2 Also there might have repeated complex conjugates For repeated real valued roots r if its multiplicity is s then the there are 3 linear independent solutions related to r 6717 tert7 7 ts7le7 t For repeated complex conjugates suppose r1 A 21 T2 A 7 211 both have multiplicity 2 The the four solutions related to r1 T2 are M cos at e sinm ten sinm t e cos at Practice Problems Section 42 22 31 undetermined coefficients method is essentially the same to Chapter 3 except that the value of s in Table 361 maybe need to be larger than 2 Practice Problems Section 43 1115

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