schaum's easy outline: differential equations (schaum's outline series)
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SCHAUMS Easy OUTLINES
B a s e d o n S c h a u m s
Out l ine o f Theory and Problems o f
Di f ferent ia l Equat ions , Second Edi t ion
b y R i c h a r d B r o n s o n , Ph.D.
A b r i d g e m e n t E d i t o r
E r i n J . B r e d e n s t e i n e r , Ph.D.
S C H A U M S O U T L I N E S E R I E SM c G R AW - H I L L
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Chapter 1 Basic Concepts and ClassifyingDifferential Equations 1
Chapter 2 Solutions of First-Order Differential Equations 8
Chapter 3 Applications of First-Order Differential Equations 20
Chapter 4 Linear Differential Equations: Theory of Solutions 29
Chapter 5 Solutions of Linear HomogeneousDifferential Equations with Constant Coefficients 33
Chapter 6 Solutions of Linear Nonhomogeneous Equations and Initial-Value Problems 39
Chapter 7 Applications of Second-Order Linear Differential Equations 47
Chapter 8 Laplace Transforms and InverseLaplace Transforms 55
Chapter 9 Solutions by Laplace Transforms 65Chapter 10 Matrices and the Matrix
Exponential 69Chapter 11 Solutions of Linear Differential
Equations with Constant Coefficients by Matrix Methods 78
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Chapter 12 Power Series Solutions 85Chapter 13 Gamma and Bessel Functions 98Chapter 14 Numerical Methods 104Chapter 15 Boundary-Value Problems
and Fourier Series 115Appendix Laplace Transforms 124Index 133
vi DIFFERENTIAL EQUATIONS
Basic Conceptsand Classifying
In This Chapter:
Differential Equations Notation Solutions Initial-Value and Boundary-Value
Problems Standard and Differential Forms Linear Equations Bernoulli Equations Homogeneous Equations Separable Equations Exact Equations
A differential equation is an equation involving an unknown function andits derivatives.
Example 1.1: The following are differential equations involving the un-known function y.
Adifferential equation is an ordinary differential equation if the unknownfunction depends on only one independent variable. If the unknown func-tion depends on two or more independent variables, the differential equa-tion is a partial differential equation. In this book we will be concernedsolely with ordinary differential equations.
Example 1.2: Equations 1.1 through 1.4 are examples of ordinary differ-ential equations, since the unknown function y depends solely on the vari-able x. Equation 1.5 is a partial differential equation, since y depends onboth the independent variables t and x.
4 5 03
dxxy+ + =(sin )
2 1+ =
dxx= +5 3
2 DIFFERENTIAL EQUATIONS
The order of a differential equation is the order ofthe highest derivative appearing in the equation.
Example 1.3: Equation 1.1 is a first-order differential equation; 1.2, 1.4,and 1.5 are second-order differential equations. (Note in 1.4 that the or-der of the highest derivative appearing in the equation is two.) Equation1.3 is a third-order differential equation.
The expressions are often used to represent, re-spectively, the first, second, third, fourth, . . ., nth derivatives of y with re-spect to the independent variable under consideration. Thus, repre-
sents if the independent variable is x, but represents
if the independent variable is p. Observe that parenthesis are used in y(n)
to distinguish it from the nth power, yn. If the independent variable istime, usually denoted by t, primes are often replaced by dots. Thus,
A solution of a differential equation in the unknown function y and theindependent variable x on the interval is a function y(x) that satisfiesthe differential equation identically for all x in .
Example 1.4: Is , where c1 and c2 are arbi-trary constants, a solution of ?
Differentiating y, we find y = 2c1cos2x 2c2 sin2x and y =. Hence, c x c x4 2 4 21 2sin cos
+ =y y4 0y x c x c x( ) sin cos= +1 22 2