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# APPLIED DIFFERENTIAL EQUATIONS MTH 256

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This 7 page Class Notes was uploaded by Miss Johan Jacobson 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 15 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

Linear ODE with Constant Coefficients Method of Undetermined Coefficients Date Feb 20 2002 Last Revision Feb 20 2002 Maple 6 Bent E Petersen bentalummitedu petersenmathorstedu Course Mth 256 Term Winter 2002 File name 2 56w2 O 02 undetermined coeffs mws For each of the following Cauchy initial value problems do the following 1 Find the complementary solution that is a general solution of the associated homogeneous equation 2 Use the method of undetermined coefficients to find a particular solution of the inhomogeneous equation 3 Combine the results of the first two parts to find a general solution of the inhomogeneous equation 4 Determine the values of the parameters arbitrary constants in the general solution found in the previous step so as to satisfy the initial values condition Does your solution agree with Maple s solution Probleml gt ode01diff yx xx 3diff yx x 2yx x 2l 32 a 2 0de0 yx 3 yx 2yxx 1 3x2 3 gt initOlyO3Dy 0l initOI 2 y0 3 Dy0 1 gt dsolveode01init01yx 9 5 2X 7x e 1 yxEx2 ExZ 4 2e Problem 2 d0239 i 3 1 X 0 e ax2yx axyx e gt ode02zdiff yx xx diff yx x lexpx fgt init02zy0lDy 02 L init02 2 y01 Dy0 2 gt dsolveode02init02yx yx xexx2eX 1 Problem3 gt ode03 diff yx xx 4yx xcos 2x xcos x 32 0de03 z yx 4 yx x cos2x x cosx x gt init03y0 ADy 0 B init03 z y0 A Dy0 B 39 gt dsolveode03init03yx 16913 2A 2 112 21 1 2 y X 2 S1I1 X COS X 8 X S1I1 X 3 X COS X X COS X S1I1X Problem 4 39 gt ode04 diff yx xx 2diff yx x 10yx expx expx sin 3 x i 3 7x Hm 0de04 yx 2 yx 10yxe e sm3x 3x2 3 gt initO4yOlDy 02 init04y01Dy0 2 7 gt dsolveode04init04yx x l e7X 1 eGX sin3 x l e7X 0033 x x eH 0033 x y 9 18 6 9 Problem 5 2 X 2 X X ix e e e 32 0de05 2yx 4yxe Bx gt init05zy0ODy 00 init05 z y0 0 Dy0 0 gt dsolveode05init05yx simplify 2x 1 72x 1 X 14 1 M i 72x e e e xe e gt ode05zdiff yx xx 4yx exp2x exp2x expx expx x 4 3 3 4 24 i yx 24e Problem 6 7 gt ode06 diff yx x3 3diff yx x2 3diff yx x yx xxexp x B3 82 3 7x 0de06 z yx3 yx 3 yx yx xx e gt init06y02DY 0l D2 Y O2 init06 z y0 2 Dy0 1 D2y0 2 7 gt dsolveode06init06yx X 3 3xe 2 7x X 4 1 7x yxx 3Ze x5e e x2 Problem7 gt ode07diff yx x4 yx xexpx expx cos x sinx 0de07 z 88 yx yx x ex e7X cosx sinx gt init07y0lDY O3 D2 Y O2 D3 Y 0l 2 3 lmt071 Y01DY03D Y02D Y01 7 gt dsolveode07init07yx simplify 7x 1 1 1 7x 1 1 3 9 1 yxEsinx EeX Zcosx2e Esinxx Zxcosx1xe x1xex Problem8 7 gt ode08zdiff yx x3 6diff yx x2 9diff yx x l4yx exp xexpx y y a X a 0de08 gym 6 g x 9 gym 14yxe e 7gt init08ylog20Dy10920D2Ylog20 2 mm me 0 Dlty1nlt2 0 D ylt1nlt2 0 7 gt dsolveode08init08yx yx 1 9x 1 7x 1 9x 78x 1 1 X 14 72x 19 7x e x e e e ln2 e e e 18 16 108 18 144 81 331776 Problem 9 gt ode09zdiff yx x2 5diff yx x 6yx x 2expx 32 E 0de09 z 2 yx 5 yx6 yx x2 ex Bx 3 gt init09y0 2Dy 0 3 imm9ya 2meX03 gt dsolve ode09 init09yx 7 X 3 X 1 2 X 2x 3 3x yx e xe x e e e 4 2 2 gt ode10diff yx x2 yx sinx COS x 32 odeIO z yx yx sinx cosx x gt init10y0 ADy 0 B mm0wmADwxmB gt dsolveodeloinit10yx simplify yx sinx sinx B cosx A sinx x x cosx T problem 10 L gt 20012002 Catalog Data Prerequisites by Topic 1 Textbook ENGR 312 Thermodynamics ENGR 312 4 credits Applications Machine and cycle processes thermodynamic relations nonreactive gas mixtures reactive mixtures thermodynamics of compressible uid ow Must be taken in order PREREQ MTH 256 CH202 for ENGR 311 Lecrec Engineering Thermodynamics ENGR 311 Math Calculus of twovariable functions Computer Use of computer to read thermo properties 93h Moran M J Shapiro H N Fundamentals ofEngmeermg Thermodynamics John Wiley amp Sons Inc New York NY second edition 1992 Course Learning Objectives By the completion of this course students are expected to 1 2 Topics bP N Schedule Prepared by A M Kanury Use the laws of Thermodynamics to analyze and design basic power and refrigeration cycles Describe the origin of thermodynamic properties and the building of property tables for any simple compressible substance Apply fundamentals of thermodynamics to problems involving either reacting e g combustion or nonreacting e g psychrometry gas mixtures Determine the equilibrium composition of a reacting mixture Calculate characteristics of subsonic and supersonic ow through nozzles and ducts with account for a normal shock if present in the ow Vapor and gas power cycles Vapor compression refrigeration Introduction to compressible ow Isentropic ow Normal shock front Maxwell relations BornTisza square Clapeyron equation of state Enthalpy internal energy and entropy differentials in terms of P v and T differentials and of cp cv Law of corresponding states Reduced enthalpy and entropy relations and charts Nonreactive mixtures Introduction to psychrometrics Mass and energy conservation in moist air processes Adiabatic saturation Wetbulb temperature Dew point Psychrometric cha1t Reactive mixtures Combustion Stoichiometry Energy conservation Enthalpy of formation of a compound Standard enthalpy of a reaction Standard enthalpy of combustion of a fuel Adiabatic ame temperature Chemical equilibrium Equilibrium constant Equilibrium composition calculation Equilibrium ame temperature and product composition Lecture Date September 2001 ENGR 312 Thermodynamics Recitation Prepared by A M Kanury Date September 2001 S L P Substantial correspondence Limited correspondence Prepared by A M Kanury Potential for correspondence instructor dependent Date September 2001 ISUMMARYISIPIPI Objective 1 Objective 2 Objective 3 Objective 4 Objective 5 Re quirem ents Course L earning Objectives ABET L a 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 d Ability to function on multidisciplinary teams 8 Ability to identify formulate and solve engineering problems Understanding of professional and ethical responsibility f g Ability to communicate effectively 11 Broad education necessary to understand the impact of engineering solutions in a global and societal context 1 Recognition ofthe need for and an ability to engage in lifelong learning 139 Knowledge of contemporary issues k Ability to use the techniques skills and modern engineering tools necessary for engineering practice 1 Abil39 to apply advanced mathematics through multivariate 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 thermal systems area including the design and real ation ofsuch systems p Ability to Work professionally in the mechanical systems area including the design and realization of such systems Student Self Assessment of Capability ENGR 312 Thermod ynamics Course Learning Objectives Mapped to ABET Goals

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