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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
For more information about this title, click here.
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
dy dt d y dt d y dt/ , / , / ,2 2 3 3and, , y y yand
d y dp2 2/d y dx2 2/
y y y y y n, , , ,...,( ) ( )4
CHAPTER 1: Basic Concepts and Classification 3
Thus, satisfies the differential equation for allvalues of x and is a solution on the interval .
Example 1.5: Determine whether is a solution of .
Note that the left side of the differential equation must be nonnegative forevery real function y(x) and any x, since it is the sum of terms raised tothe second and fourth powers, while the right side of the equation is neg-ative. Since no function y(x) will satisfy this equation, the given differ-ential equation has no solutions.
We see that some differential equations have infinitely many solu-tions (Example 1.4), whereas other differential equations have no solu-tions (Example 1.5). It is also possible that a differential equation has ex-actly one solution. Consider , which for reasons identicalto those given in Example 1.5 has only one solution .
You Need to Know
A particular solution of a differential equation is anyone solution. The general solution of a differentialequation is the set of all solutions.
Example 1.6: The general solution to the differential equation in Ex-ample 1.4 can be shown to be (see Chapters Four and Five)
. That is, every particular solution of the differentialequation has this general form. A few particular solutions are: (a)
(choose and ), (b) (chooseand ), and (c) (choose ).c c1 2 0= =y 0c2 0=c1 1=
y x= sin 2c2 3= c1 5=x x5 2 3 2sin cosy =
c x c x+1 22 2sin cosy =
y 0( ) + =y y4 2 0
= 1( ) +y y4 2y x= 2 1
( , ) y c x c x= +1 22 2sin cos
+ = + += + + +=
y y c x c x c x c x
c c x c c x
4 4 2 4 2 4 2 2
4 4 2 4 4 2
1 2 1 2
1 1 2 2
( sin cos ) ( sin cos )
( )sin ( )cos
4 DIFFERENTIAL EQUATIONS
The general solution of a differential equation cannot always be ex-pressed by a single formula. As an example consider the differential equa-
tion , which has two particular solutions and .
Initial-Value and Boundary-Value Problems
A differential equation along with subsidiaryconditions on the unknown function and its de-rivatives, all given at the same value of the in-dependent variable, constitutes an initial-valueproblem. The subsidiary conditions are initialconditions. If the subsidiary conditions are giv-en at more than one value of the independentvariable, the problem is a boundary-value prob-lem and the conditions are boundary conditions.
Example 1.7: The problem is an initialvalue problem, because the two subsidiary conditions are both given at
. The problem is a boundary-valueproblem, because the two subsidiary conditions are given at x = 0 and x = 1.
A solution to an initial-value or boundary-value problem is a func-tion y(x) that both solves the differential equation and satisfies all givensubsidiary conditions.
Standard and Differential Forms
Standard form for a first-order differential equation in the unknown func-tion y(x) is
where the derivative appears only on the left side of 1.6. Many, butnot all, first-order differential equations can be written in standard formby algebraically solving for and then setting f(x,y) equal to the rightside of the resulting equation.
=y f x y( , )
+ = = =y y e y yx2 0 1 1 1; ( ) , ( )x = p
+ = = =y y e y yx2 1 2; ( ) , ( )p p
y 0y x= 1 / + =y y2 0
CHAPTER 1: Basic Concepts and Classification 5
The right side of 1.6 can always be written as a quotient of two oth-er functions M(x,y) and N(x,y). Then 1.6 becomes
which is equivalent to the differential form
Consider a differential equation in standard form 1.6. If f(x,y) can be writ-ten as (that is, as a function of x times y, plus an-other function of x), the differential equation is linear. First-order lineardifferential equations can always be expressed as
Linear equations are solved in Chapter Two.
A Bernoulli differential equation is an equation of the form
where n denotes a real number. When n = 1 or n = 0, a Bernoulli equationreduces to a linear equation. Bernoulli equations are solved in ChapterTwo.
A differential equation in standard form (1.6) is homogeneous if
for every real number t. Homogeneous equations are solved in ChapterTwo.
f tx ty f x y( , ) ( , )=
+ =y p x y q x yn( ) ( )
+ =y p x y q x( ) ( )
f x y p x y q x( , ) ( ) ( )= +
M x y dx N x y dy( , ) ( , )+ = 0
N x y( , ),dy dx M x y/ ( , ) /=
6 DIFFERENTIAL EQUATIONS
In the general framework of differential equations,the word homogeneous has an entirely differentmeaning (see Chapter Four). Only in the context of first-order differential equations does homoge-neous have the meaning defined above.
Consider a differential equation in differential form (1.7). If M(x,y) = A(x)(a function only of x) and N(x,y) = B(y) (a function only of y), the differ-ential equation is separable, or has its variables separated. Separableequations are solved in Chapter Two.
A differential equation in differential form (1.7) is exact if
Exact equations are solved in Chapter Two (where a more precise defini-tion of exactness is given).
M x y
N x y
( , ) ( , )
CHAPTER 1: Basic Concepts and Classification 7
In This Chapter:
Separable Equations Homogeneous Equations Exact Equations Linear Equations Bernoulli Equations Solved Problems
The solution to the first-order separable differential equation (see Chap-ter One).
(2.1)A x dx B y dy( ) ( )+ = 0
where c represents an arbitrary constant.(See Problem 2.1)The integrals obtained in Equation 2.2 may be, for all practical pur-
poses, impossible to evaluate. In such case, numerical techniques (seeChapter 14) are used to obtain an approximate solution. Even if the indi-cated integrations in 2.2 can be performed, it may not be algebraicallypossible to solve for y explicitly in terms of x. In that case, the solution isleft in implicit form.
Solutions to the Ini...