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# APPLIED DIFFERENTIAL EQUATIONS MTH 256

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This 11 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 256 at Oregon State University taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/224447/mth-256-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15

MTH 256 APPLIED DIFFERENTIAL EQUATIONS 20012002 Catalog Data Prerequisites by Topic Textbook MTH 256 4 credits First order linear and nonlinear equations and second order linear equations Applications to electric circuits and mechanical oscillators Introduction to the Laplace transform and higher order equations Introduction to linear systems of differential equations eigenvalues and normal modes Related matrix and linear algebra concepts Solution methods and applications appropriate for science and engineering PREREQ MTH Note The material on matrix and linear algebra topics is now covered in MTH 306 and systems of DEs are not covered in MTH 256 The catalog will be updated next year 1 Ability to handle standard algebraic trigonometric and vector calculations at the college level 2 Proficiency with differentiation and integration of functions of one two and three variables including the chain rule William Boyce amp Richard Diprima Elementary Differential Equa trans and Boundary Value Problems Wiley 7th edition 2000 Course Learning Objectives By the completion of this course students are expected to Prepared by J W Lee 1 Identify and solve first order differential equations that are separable exact homogeneous or linear or can be reduced to such equations by a simple change of variable Construct and analyze models for physical systems such as for mixing cooling radioactive decay that can be described by first order linear or nonlinear differential equations Also solve nonlinear autonomous second order differential equations and apply these methods to some physical problems Understand the basic structure of the solution space for linear differential equations principally of second order and use this structure to solve such equations Construct and analyze models for physical systems such as for small mechanical vibrations and electric circuits that can be described by second order linear differential equations Use Laplace transforms to solve initial value problems 1 Introduction to differential equations and the basic existence and uniqueness theorem for initial value problems 2 classes 2 Solution methods for first order linear and nonlinear differential equations and applications 5 classes Date February 2002 MTH 256 APPLIED DIFFERENTIAL EQUATIONS 3 Schedule Structure of the solution space for linear differential equations linear dependence and independence of solutions superposition principle Wronskian 3 classes Solution methods characteristic equation undetermined coef cients variation of parameters for second order homogeneous and nonhomogeneous constant coef cient equations 7 classes Applications to small vibrations of mechanical systems and to electric circuits 3 classes Laplace transforms and solution of initial value problems 5 classes Catch up and review 2 classes Lecture 1 hour three times per week Recitation 1 hour each week Prepared by J W Lee Date February 2002 Ob active 1 Ob active 2 Ob active 3 Ob active 4 Ob active 5 S P Re quirem ants Course L earning Obj activ a s ABET S a ISUMMARY s Ability to apply math science and engineering b Ability to design and conduct experiments as Well as to analyze and interpret data 0 Ability to design a system component or process to meet desired needs Substantial correspondence Prepared by J W Lee d Ability to function on multidisciplinary teams 8 Ability to identify formulate and solve engineering problems Understanding ofprofessional and ethical responsibility f g Ability to communicate effectively Potential for correspondence instructor dependent 11 Broad education necessary to understand the impact of engineering solutions in aglobal and societal context 1 Recognition of the need for and an ability to engage in lifelong learning 139 Knowledge of contemporary issues k Abil39ty to use the techniques skills and modem engineering tools necessary for engineering practice 1 to apply advanced mathematics through mult ariate calculus and differential equations m Familiarity with statistics and linear algebra n Knowledge of chemistry and calculusbased physics with depth in at least one 0 Ability to Work professionally in the themial systems area inclu ing the design and reali ation of such systems Date February 2002 p Abil to Work professionally in the mechanical systems area including the design and realization of such systems Student Self Assessment of Capability Course Learning Objectives Mapped to ABET Goals MTH 256 APPLIED DIFFERENTIAL EQUATIONS Weeks 13 Review Fall 2006 Mth 256 Bent E Petersen This review deals with recipes for solving rst order ordinary di erential equations as studied in the rst three weeks of Mth 256 Fall 2006 The theoretical parts of the course are not reviewed here The problems listed here do not represent every type of problem we discussed Answers indicated below were found mostly by Maple for fun and so may diiTer in some inessen tial way from the answers that you nd While I copy and paste the ETEX version of the answers as provided by Maple I can t help but clean up Maple s ETEX expressions a bit Undoubtably I introduced a few errors so you should rely more on yourself than on the provided answers Enjoy 1 Picard Iteration Problem 1 The initial value problem dy 2 2 7 t 0 1 dt y y can be solved in terms of certain Bessel functions If we expand the solution in a Taylor series we Obtain 4 7 6 37 404 369 t 1 t t2 43 44 45 it6 7t7 it8 m y 3 6 5 30 315 280 If you perform three Picard iterations starting with yo I 1 what do you obtain How does your result compare with the solution given above Answer 4 5 8 29 47 1 t t2 43 44 7t5 it6 7t7 m 3 6 15 90 315 2 Linear First Order ODE Problem 2 Solve the initial value problem tantytant y0 3 M th 256 Weeks 1 3 Review Answer yt 12 cos I Problem 3 Solve the initial value problem El s1nt y7r2 Answer t 7 t 1 7 ya s1n cos Problem 4 d l 7 L 2 dt 7 3 H Answer 2 yt t 73 3t 9 logt 73 Cgt Problem 5 dy E cotty s1nt Answer yltrgt r 0 sinltrgt Problem 6 d y 7 J E i t s1n logt Answer yt it coslogt Ct Problem 7 A certain radioactive substance decays to 85 of its original mass in 36 hours Find the halfilife Answer 15354 hours Bent E Petersen Page 2 Fall 2006 M th 256 Weeks 1 3 Review 3 Newton s Law of Cooling Problem 8 A cup of hot coffee initially at temperature 1340 F is brought into a room of temperature A the ambient The colTee begins to cool down and of course the room warms up a bit However the heat capacity of the room is so large compared to the cup of colTee that we may assumeA remains constant After 2 minutes the coffee is observed to have the temperature 1000 K Another minute later the colTee is observed to have the temperature 900 F Deduce the temperature of the room Answer A 65640F Problem 9 A cup of colTee initially at 1900 F is brought into aroom at 650 F After 2 minutes the temperature of the colTee is 1450 F Predict the temperature of the colTee an additional minute later Answer 1290 F Problem 10 Consider an insulated box with internal temperature T Assume that the ambi ent external temperature A is changing linearly for a while at least say A A0 A1 t where A0 andA1 are constants and t is time According to Netwon s law of cooling we have dT 7 7k TiA dt where k is a constant depending on the insulation of the box Find the temperature Tt in terms of t A0 A1 and k Do not neglect the arbitrary constant Answer 1 Tt Ao A1ltt7 ggt Ce quot Bent E Petersen Page 3 Fall 2006 M th 256 Weeks 1 3 Review 4 Mixing Problems Problem 11 A brine solution consisting of 006 ozgal salt dissolved in water ows into a large tank at the rate 30 galmin The solution inside the tank is kept wellmixed and ows out of the tank at the rate 20 galmin Ifthe tank initially contains 500 gal of brine of concentration 003 02 gal determine the amount of salt in the tank after t minutes When will the concentration of salt in the tank reach 005 02 gal Assume the tank is so large that it does not over ow Answer 221 min Problem 12 A 200 L tank initially contains 100 L of brine of concentration 3 g L salt ie 3 grams salt per liter water Brine of concentration 5 gL salt runs into the tank at 8 Lmin The wellimixed solution is drawn off at the rate 6 Lmin Find the concentration of salt in the solution in the tank at the moment that the tank begins to over ow Answer 4875 gL Problem 13 A 100 gal tank initially contains 20 gal of brine of concentration 024 ozgal salt Brine of concentration 0 l 8 02 gal ows into the tank at 3 galmin and the wellmixed solution is drawn off at the rate of l galmin Find the amount of salt in the tank at the very moment that it begins to over ow Answer 18537 oz 5 Separable First Order ODE Problem 14 For a body of mass m say a person falling near the surface of the earth we may assume the acceleration of gravity g is a constant Also in many cases we may assume the drag is proportional to the square of the velocity Thus the equation of motion is Bent E Petersen Page 4 Fall 2006 M th 256 Weeks 1 3 Review Here v is the downward velocity If we reparametrize this equation by the distance y fallen try it we obtain m v Q mg 7 kv2 dy Assuming the body falls from rest nd the velocity in terms of the distance fallen From your solution determine the terminal ultimate or limiting velocity symbolically if you fall a very long way How far does a 60 kg person have to fall to reach 95 of the terminal velocity if k 060 kgm Note k depends on the person s aspect For feet or head rst we may have roughly k 006 Note you do not need the acceleration of gravity g to answer the questions above but you do to compute the actual speed Use g 980 msec2 to compute the 95 of terminal velocity achieved by our intrepid aeronaut Answ ers 12 12 e lt1 7 mg 12 7 gt and 1164 m with a speed of 297 msec or about 665 mph Problem 15 Solve the initial value problem dy xyy i 1 7 dx x Answer y ixexi1 Problem 16 Solve the initial value problem dp E ez p 7p 100 05 Find lim pt Answer 1 W W The limit is 1 Bent E Petersen Page 5 Fall 2006 Mth 256 Weeks 1 3 Review 6 Bernoulli ODE Problem 17 Answer Problem 18 Answer Solve the ordinary di erential equation 3 yjtanx ydx x x 39 x4y4 4 logcosx C Solve the ordinary di erential equation 61 x67 2y logx 0 1 1 C 1 7 l z 0gx 2 x 7 Homogeneous First Order ODE and related substitutions Problem 19 homogeneous Answer Problem 20 Answer Use the substitution y x2w to solve the ordinary di erential equation not Q 7 2y2 x3 dx xy y22x3Cx40 Solve the initial value problem dy 7 4x2 y2 dx i 2xy y2 2 yx 4x2 7 6x 12 Bent E Petersen Page 6 Fall 2006 M th 256 Weeks 1 3 Review Problem 21 Solve the initial value problem g7ylt1x2gt7lt1xzx yltogt2 Answer 3 yx l exxT You can also solve this equation as a linear ode 8 Orthogonal Trajectories Problem 22 Find the family of orthogonal trajectories to the oneparameter family of hy perbolas given by 2y2 7 x2 06 Answer y C x2 Problem 23 Find the family of orthogonal trajectories to the oneparameter family of cubics y 06x Here 06 is the parameter that is an arbitrary constant Answer 3 x2 7 I3 Here 3 is a parameter so we see we get a family of ellipses Problem 24 Consider the lparameter family of hyperbolas and ellipses given by x2 7 Gay2 1 Here 06 is the parameter Find the lparameter family of orthogonal trajectories Answer x2 y2 7 210gx L3 BentE Petersen Page 7 Fall 2006 M th 256 Weeks 1 3 Review 9 Integrating Factors and Exact ODE Problem 25 For what values of p and q is xpyq an intgrating factor for the ordinary dilTer ential equation 6y2 3y7 4xy dx 73x2 3x 8xyaly 0 Answer p 2 q 2 Note Maple nds the integrating factor it It should be possible to encourage Maple to also nd it x2y2 but I haven t had any luck Problem 26 Solve the exact ordinary differential equation al 2xy3 7y2 7 2 3x2y2 7 2xy3diy 0 x Answer xzy3 ixyz 72x3yC To get this answer in Maple you have to tell Maple you want an implicit solution Otherwise Maple solves this cubic equation for y which leads to quite a mess So use a command like alsolveoale implicit If you have used implicit as a variable name you ll have to escape it using single quotes to prevent evaluation and confusion Problem 27 Find an integrating factor which depends only on y and then solve the differen tial equation dy 2yy2 6xy 4x3xyi6x2E 0 Answer y Copyright 2006 Bent E Petersen This document may be freely alu plicateal unaltered If you make changes even improvements please remove my name Bent E Petersen 24 hour phone numbers Department of Mathematics of ce 541 7375163 Oregon State University Corvallis OR 973314605 fax 541 7370517 email petersenmathoregonstateedu httporegonstateeduNpeterseb Bent E Petersen Page 8 Fall 2006

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