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Differential Equations

by: Ashley Ernser II

Differential Equations Math 315

Ashley Ernser II
GPA 3.95

Kevin Vixie

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Kevin Vixie
Class Notes
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This 2 page Class Notes was uploaded by Ashley Ernser II on Thursday September 17, 2015. The Class Notes belongs to Math 315 at Washington State University taught by Kevin Vixie in Fall. Since its upload, it has received 11 views. For similar materials see /class/205964/math-315-washington-state-university in Mathematics (M) at Washington State University.


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Date Created: 09/17/15
Math 315 Study Guide for Test 1 Summer 2011 Chapter 1 Introduction to Differential Equations This chapter is a review of basic differential and integral calculus and the ideas are used extensively throughout the text However no questions will come directly from this chapter Chapter 2 FirstOrder Differential Equations 1 Equations of the form 31 f t y can be linear or nonlinear depends on f What are de nition of a solution general solution and intervals of existence Separable equations 31 f t y gthy easy to solve assuming the integration is not daunting Understand that solutions can be explicit or implicit Models of motion uses Newton39s laws of motion to provide a physical example of rstorder equations You will not be asked to derive a model beyond the difficulty of say 6 p45 Linear equations main solution technique is construction of an integrating factor The authors introduce the terms homogeneous and particular solutions These terms were not emphasized during lecture You will hear them repeatedly when we discuss Chapter 4 No variation of parameters We39ll see plenty of it in Chapter 4 Mixing problems you are guaranteed to see a mixing problem of some form It will be one where the integration and algebra of the problem should be straightforward Tip let dimension and units guide you Exact differential equation know how to construct the function F where dF 0 is equal to the differential equation we are interested in solving Since constructing an integrating factor N W P 9quot 9 for nonlinear equations can be time consuming they are not likely to appear on the test Know what homogeneous functions and homogeneous equations are and how to solve them The transformation yx vxx transforms the equation into a separable one Here I am using the term homogeneous in the context of multiplicative scaling ie f tx ty t f x y Existence and uniqueness know Theorems 76 and 716 and understand questions such as those on p 86 1 10 and p 87 25 32 Dependence on initial conditions nothing from this section will appear on the test Autonomous equations and stability know what is meant by equilibrium points stability of equilibrium points phase line direction eld and sketching solutions using the qualitative methods in this section 090 Chapter 3 Modeling and Applications Material from Section 31 will show up on the test Know how to use and adapt Malthusian and logistics models eg how to introduce a harvesting term into the model Know how to write a model in dimensionless form This is discussed on p 42 43 and I discuss the idea in this section Appendix Complex Numbers and Matrices p 699 702 The appendix is a review of some basic concepts involving complex algebra You should know how to add subtract multiply and divide complex numbers how to find magnitude or absolute value know complex conjugate polar form of complex numbers and the very important Euler39s formula Know proposition A11 Hopefully you tried all of the suggested problems and more Practice test Please note that this practice test is meant to give you a feel for the length and difficulty of the test you will take You will not necessarily see problems like this on the test you take so please do not come to me and complain that the sample was nothing like the actual test 1 Do the following a Find the real 1 b such that a bi b Find the real 1 9 such that 1 i re 0 Write ft 2 cos4t 2 sin4t in the formA cos wt p 2 Find general solution to 2xy2x a39 y X21 b xydx y xdy 0 3 A tank initially contains 100 gal of water in which is dissolved 2 lbs of salt The saltwater solution containing 1 lb of salt for every 4 gal of solution enters the tank t a rate of 5 gal per minute The solution leaves the tank at the same rate allowing for a constant solution volume in the tank Find a model for the concentration of salt in the tank at any time t 4 Suppose that x is a solution to the initial value problem 96 Xa X with 360 12 Show 1t2x2 that 0 lt xt lt 1 for all t which x is de ned 5 Adjust the standard logistic equation to re ect the fact that a fixed percentage y of the population is harvested per unit time Use qualitative analysis to discuss the fate of the population In your analysis discuss two particular cases 1 y lt r and 2 y gt r 6 Giveny y2 3y 4 a Determine the equilibrium solutions Determine whether your solutions in a are stable or unstable Sketch the phase line The equilibrium solutions divide the ty plane into regions Sketch at least one solution trajectory in each region 99


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