Problems in Differential Equations

J. L. Brenner

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486489421
• eBook ISBN: 9780486782829
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

A supplement for elementary and intermediate courses in differential equations, this text features more than 900 problems and answers. Suitable for undergraduate students of mathematics, engineering, and physics, this volume also represents a helpful tool for professionals wishing to brush up on their problem-solving skills.
The book is divided into twenty sections, each preceded by a clear and logical explanation of the basic ideas needed for solving the problems within the section. Many fully explained illustrative problems appear throughout the text. Subjects include applied routine and nonroutine problems in vibrations, electrical engineering, mechanics, and physics. Stars indicate advanced problems. Short mathematical and numerical tables are provided at the end of the book.

Section 1

ISOCLINES. CONSTRUCTION OF THE DIFFERENTIAL EQUATION FOR A FAMILY OF CURVES ISOGONAL TRAJECTORIES

If the differential equation y′ = f(x, y), is satisfied by a curve going through the point (x, y), then the tangent line to the curve at the point in question must have slope y′, that is the angle α which this line makes with the x axis must satisfy α = arctan f(x, y). The geometric locus of those points for which y′ = f(x, y) has a constant value k is called an isocline. Thus the equations of the isoclines are f(x, y) = k, where k is constant on each isocline.

To solve the differential equation y′ = f(x, y) geometrically, it is sufficient to draw a number of isoclines and to sketch a curve which crosses each isocline with the correct slope. Examples of this construction are given in any elementary text on differential equations.

The loci which intersect every curve of a given family of curves at a constant preassigned angle ϕ are called isogonal trajectories. The angle β which the trajectory makes with the x axis is thus ϕ units greater or less than the angle α which the intersected trajectory makes with the same axis: β = α ± ϕ. Suppose

is the differential equation corresponding to a particular family of curves and suppose

is the differential equation of a family of isogonal trajectories. In other words, tan α = f(x, y), tan β = f₁(x, y). Thus when equation (1) and the angle ϕ are given it is easy to write down the formula for tan β, and thus to set up the differential equation (2) from which the isogonal trajectories ...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• Contents
• Introduction
• 1 Isoclines, Construction of the Differential Equation for a Family of Curves Isogonal Trajectories
• 2 Equations in which the Variables are Separable.
• 3 Geometrical and Physical Problems
• 4 Homogeneous Equations
• 5 Linear First Order Equations
• 6 Exact Equations
• 7 Questions on Existence and Uniqueness of Solutions
• 8 Equations in which the Derivative Appears Implicitly
• 9 Miscellaneous First Order Equations
• 10 Equations which can be Reduced to Equations of Lower Order
• 11 Linear Equations with Constant Coefficients
• 12 Linear Equations with Variable Coefficients
• 13 Series Development of Solutions of Equations
• 14 Linear Systems with Constant Coefficients
• 15 Lyapunov Stability
• 16 Singular Points
• 17 Problems in the Theory of Oscillations
• 18 Dependence of Solutions on Initial Conditions and on Parameters. Approximate Solution of Differential Equations
• 19 Non-Linear Systems
• 20 First Order Partial Differential Equations
• Answers
• Miscellaneous Formulas
• Derivatives
• Some Integrals
• Tables: Square Roots, Cube Roots