MATH529

SP.22: Homework 1

Instructions

This homework assignment is a review of ordinary differential equation theory and solution techniques. It is also meant as the first step in learning to use computational software systems (i.e., Mathematica) to present and verify problem solutions.

Read Chapter 1 of Zill’s textbook (attached) to recall basic concepts. Solve as many exercises as needed to re-acquaint yourself with solution techniques. Though only a limited number of exercise and problem solutions are assigned as formal homework, you may need to work through more (perhaps, many more) problems by yourself to gain mastery of solution techniques.

As stated on course website, 4 course points are attributed to each homework assignment with exercises valued at 0.25 points, problems at 0.5 points, and projects at 1 point. You are free to choose among suggested exercises. This is meant to allow students to self-assess and build proficiency. Mastery of the techniques of mathematical physics is a joint endeavor between students and instructors, and you should strive to especially attempt problems that seem difficult at your current level of understanding so that feedback from grading can advance comprehension.

Model solutions are included in all homework assignments to demonstrate solution techniques and level of detail required in drafting a solution.

Mathematica notebooks use a hierarchical organization of cells: ALT+1 is a title cell, ALT+2 a subtitle, ALT+4 a section, ALT+5 a subsection, etc. ALT+7 is a text cell, ALT+9 is a computational input cell.

When drafting a solution, start with a brief statement of solution technique. Introduce intermediate notations as needed to simplify presentation of solution.  Conclude with a brief statement, e.g., “solution is verified”

In Mathematica 13, CTRL+SHIFT+4 with Enable Dynamics option turned on introduces a LATEX equation field. Use a search engine for common LATEX commands, e.g. {\dfrac{a}{b}} produces a fraction
, and \cos x produces .

Exercises

From p.11: 1-10

Model solution:  1.1.6 (p. 11)

Statement

State order and linearity of .

Solution

Equation is of second order and nonlinear.

Problems

From pp.11-12: 13-20, 23-26

Model solution:  1.1.16 (p. 11)

Statement

Verify that  is a solution of .

Solution

Compute derivatives, and substitute into DE.

y[x_]= -Cos[x] Log[Sec[x]+Tan[x]]

DE = y''[x]+y[x] == Tan[x]

Simplify[DE]

Solution is verified.