Ordinary Differential Equations Homework Solutions

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The first definition that we should cover should be that of differential equation. A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives.

The order of a differential equation is the largest derivative present in the differential equation. In the differential equations listed above \(\eqref{eq:eq3}\) is a first order differential equation, \(\eqref{eq:eq4}\), \(\eqref{eq:eq5}\), \(\eqref{eq:eq6}\), \(\eqref{eq:eq8}\), and \(\eqref{eq:eq9}\) are second order differential equations, \(\eqref{eq:eq10}\) is a third order differential equation and \(\eqref{eq:eq7}\) is a fourth order differential equation.

A solution to a differential equation on an interval \(\alpha < t < \beta \) is any function \(y\left( t \right)\) which satisfies the differential equation in question on the interval \(\alpha < t < \beta \). It is important to note that solutions are often accompanied by intervals and these intervals can impart some important information about the solution. Consider the following example.

So, we saw in the last example that even though a function may symbolically satisfy a differential equation, because of certain restrictions brought about by the solution we cannot use all values of the independent variable and hence, must make a restriction on the independent variable. This will be the case with many solutions to differential equations.

[t,y] =ode45(odefun,tspan,y0),where tspan = [t0 tf], integrates the system ofdifferential equations y'=f(t,y) from t0 to tf withinitial conditions y0. Each row in the solutionarray y corresponds to a value returned in columnvector t.

All MATLAB® ODE solvers can solve systems of equations ofthe form y'=f(t,y),or problems that involve a mass matrix, M(t,y)y'=f(t,y).The solvers all use similar syntaxes. The ode23s solveronly can solve problems with a mass matrix if the mass matrix is constant. ode15s and ode23t cansolve problems with a mass matrix that is singular, known as differential-algebraicequations (DAEs). Specify the mass matrix using the Mass optionof odeset.

(12/7) A review sheet hasbeen posted (see also section "Review for the Final Exam").(12/6) If you liked this class and want to learn aboutsystems of differential equations, nonlinear equations, chaos, andrelated things, you should take my Math 134: Dynamical Systemsclass which I plan to teach in the Fall of 2005.New Office HoursWeek 12/6-12/10: M & W regular o.h.; F 8:30-10 & 1:30-3Week 12/13-12/17: M 10-11; Tu & Th 8:30-9:30Review for the Final ExamThe final is on: December 14, 12:15-2:30, in MH 320 forSection 04 (the 12:30 section), December 16, 12:15-2:30, in MH424 for Section 05 (the 1:30 section). Note that both exams are inour usual classrooms.There will be six problems.You are allowed to bring a 3X5" note card to the exam.To review, go over the quizzes and the midterms,as well as the these problems:Ex. 1.2: #12, 16, 13; Ex. 1.3: #6; Review exercies for Ch. 2: #1, 3, 7, 9, 17, 23, 28, 31, 32, 40; Review exercies for Ch. 4: #5, 13, 21, 23, 24, 25, 30, 31, 32,33, 35;Review exercies for Ch. 7: #1, 5, 6, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29.Check out Math 133A essentials.Some useful trigonometric identities.Sample final no.1: (1) Quiz 3; (2) Quiz 7; (3) Midterm2, #1; (4) Midterm 2, #3; (5) Ch. 7 review ex., #17; (6) Ch. 7 reviewex., #23.Sample final no.2(12/1) Solutions to Quiz 9 have beenposted.(11/30) Homework 12 has beenposted. You should do it by December 8, but you do not need to turn itin.(11/28) My afternoon office hour tomorrow, November29, will be only a half hour, from 2:30 to 3.(11/18) Solutions to Quiz 8 have beenposted.(11/16) Homework 11 has been postedand is due on November 29. Quiz 9 will be on December 1.(11/14) Quiz 8 will be on Wednesday, November 17.(11/9) Homework 10 has been posted andis due on November 17. Quiz 8 will be that same day orthe Friday after that - I'll let you know.(11/8) Solutions to Midterm 2 havebeen posted.(11/7) On the midterm, you are allowed too bring a 3 X5" note card.(11/2) As I announced in class, there will be no quiz onWednesday, 11/3. So today, get out and vote! You can find yourpolling place here: (10/27) Solutions to Quiz 7 have beenposted.(10/26) Homework 9 has been posted and isdue on November 3, except for Chapter 4 review problems whichshould be done before 11/8.(10/20) Solutions to Quiz 6 have beenposted.(10/19) You can now send me anonymousfeedback. Please take a moment to let me know what you think about theclass.(10/19) A new How to... section hasbeed added.(10/19) Homework 8 has been posted and isdue on October 27. Quiz 7 will be on the same day.(10/14) Solutions to Quiz 5 have beenposted.(10/12) Homework 7 has been posted and isdue on October 20. Quiz 6 will be on the same day.(10/7) Homework 6 has been posted and is dueon October 13. Quiz 5 will be on the same day.(10/4) Solutions to Midterm 1 havebeen posted below.(10/3) On the midterm, you are allowed to bring a 3 X 5"card with notes.(9/27) Solutions to Quiz 4 have beenposted.(9/27) An (electronic) handout about the brachistochrone problem has been posted. Imentioned it in the 12:30 class on 9/24 as part of an anecdote aboutNewton.(9/26) Homework 5 has been posted and isdue on October 4. Midterm 1 will be on the same day and willcover homework assignments 1-4.(9/21) Solutions to Quiz 3 have beenposted.(9/19) Homework 4 has been posted. It'sdue on September 27. Quiz 4 will be on the same day.(9/15) Last day to add is September 21.(9/13) Solutions to Quiz 2 have been posted.(9/12) Homework 3 has been posted. It isdue on September 20. Quiz 3 will be on the same day.(9/8) I just noticed that Section 2.2 is part of the secondhomework and that I haven't covered it yet. I therefore move the duedate for Homework 2 to Monday, September 13. Quiz 2 will be onthe same day (i.e., Monday).(9/8) Starting Monday, 9/13, my morning office hour MWFwill move from 10:30-11:30 to 8:30-9:30.(9/7) I have to cancel my 2:30 office hour on Wednesday,September 8.(9/6) A handout on basic integralsand methods of integration I expect you to know has been posted.(9/6) Solutions to Quiz 1 have been posted.(8/31) Homework 2 has been posted and isdue on September 10. Quiz 2 will be on the same day.(8/27) There will be a lottery on Monday todetermine who gets to add the class. The number of slots depends onenrollment on Monday morning. The time and place of the lottery willbe announced in class.(8/26) Starting Friday, August 27, the 1:30 section will bemeeting in MH 424. (8/25) Homework 1 has been posted and isdue on Wednesday, September 1.(8/25) Quiz 1 will be on Friday, September 3.

SyllabusFirst order equations, higher order linear equations,applications, Laplace transforms, series solutions. Additional topics.We will cover most or parts of Chapters 1, 2, 4, 7, and 8.Green sheet

Mathematical models of phenomena in which a rate of change of the state variables is specified lead to formulation of differential equations. Such models arise throughout the physical, biological, and social sciences. This introductory course presents the theoretical framework for differential equations, reinforced by numerous applications and use of software systems.

Homework generally consists of exercises from the textbook. Homework is graded by TA's. Pay particular attention to the homework solutions to learn how to succintly and correctly present mathematical answers. 2b1af7f3a8