Differential Equations

Robert P. Gilbert, George C. Hsiao, Robert J. Ronkese

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000402575
• Edition: 2
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

This book illustrates how MAPLE™ can be used to supplement a standard, elementary text in ordinary and partial differential equation. The authors are firm believers in the teaching of mathematics as an experimental science where the student does numerous calculations and then synthesizes these experiments into a general theory. The goal of the book is to teach the students enough about the computer algebra system MAPLE™ so that it can be used in an investigative way. This book was developed through ten years of instruction in the differential equations course.

Chapter 1

Introduction to the Maple DEtools

The purpose of this chapter is to introduce some Maple features specially related to differential equations. In some instances Maple can find an analytical expression for the solution of an ordinary differential equation (ODE). For this purpose we need to introduce certain syntax which will be given in this chapter.

1.1 Analytical Solutions and Their Plotting

Let us consider the first-order differential equation,

$$\begin{array}{l}\frac{dy(x)}{dx}=f(x,y).\hfill \end{array}$$(E)

To obtain the general solution for a particular form of the above equation, we first load DEtools as follows:

> with(DEtools):

Let us chose the right-hand side of the above equation ¹ as

> f:=2*x*y(x)+x;

$$\begin{array}{l}f:=2xy(x)+x\hfill \end{array}$$

The syntax for entering and solving the differential equation with this right-hand-side are then given as

> deq:=diff(y(x),x)=f;

$$\begin{array}{l}deq:=\frac{\partial }{\partial x}\text{y}(x)=2xy(x)+x\hfill \end{array}$$

> gsoln:=dsolve(deq,y(x));

$$\begin{array}{l}gsoln:=\text{y}(x)=-\frac{1}{2}+e^{(x^2)}\_C1\hfill \end{array}$$

__________________________

¹Notice that Maple requires the unknown function to be inputted as $y(x)$

To plot the solutions all parameters must be given as floating point numerical values. First we must load plots. Notice that the use of the colon at the end of the input line suppresses the output from this line.

> with(plots):

For ODEs with side conditions, such as an initial value condition, abreviated as (IVC), one first defines the condition as

> ic:=y(0)=y0;

$$\begin{array}{l}ic:=\text{y}(0)=y0\hfill \end{array}$$

> soln:=dsolve(deq...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• 1 Introduction to the Maple DEtools
• 2 First-Order Differential Equations
• 3 Numerical Methods for First-Order Equations
• 4 Differential Equations with Constant Coefficients
• 5 Applications of Second-Order Linear Equations
• 6 Two-Point Boundary Value Problems, Catalytic Reactors and Boundary-Layer Phenomena
• 7 Eigenvalue Problems
• 8 Power Series Methods for Solving Differential Equations
• 9 Nonlinear Autonomous Systems
• 10 Integral Transforms
• 11 Partial Differential Equations
• 12 Transmutations
• Bibliography
• Index