Author(s): Abraham Cohen
Publisher: D.C. Heath & Company
Language: English
Pages: 271
Front Cover
CHAPTER
DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS SECTION PAGE 1 Differential Equation Ordinary and Partial Order, Degree
Solution of an Equation ·
Derivation of a Differential Equation from its Primitive
General, Particular Solution
CHAPTER II
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND FIRST DEGREE 5 Exact Differential Equation Integrating Factor •
General Plan of Solution •
Condition that Equation be Exact
Exact Differential Equations II
Variables Separated or Separable
Homogeneous Equations
Equations in which M and N are Linear but not Homogeneous
Equations of the Form yfi(xy)dx + xf2(xy)dy = 0
Linear Equations of the First Order
Equations Reducible to Linear Equations
Equations of the Form x'yº (my dx+nx dy)+x°y³ (µy dx+vx dy)=0
Integrating Factors by Inspection
Other Forms for which Integrating Factors can be Found
Transformation of Variables ·
Summary
CHAPTER III
Differential Equation of a Family of Curves
Geometrical Problems involving the Solution of Differential Equations
Orthogonal Trajectories
Physical Problems giving Rise to Differential Equations
CHAPTER IV
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND HIGHER DEGREE THAN THE FIRST 24 Equations Solvable for
Equations Solvable for y 26 Equations Solvable for x
Clairaut's Equation
Summary
CHAPTER V
SINGULAR SOLUTIONS 29 Envelopes 30 Singular Solutions
Discriminant
Singular Solution Obtained directly from the Differential Equation
Extraneous Loci
Summary PAGE 38 43 49 58 ii cu 52 55 56
69
CHAPTER VI
TOTAL DIFFERENTIAL EQUATIONS
Total Differential Equations
Method of Solution 76
Homogeneous Equations
Equations involving more than Three Variables
Equations which do not satisfy the Condition for Integrability
Geometrical Interpretation 84
Summary
CHAPTER VII
LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 42 General Linear Differential Equation
Linear Differential Equations with Constant Coefficients Complemen-
SECTION PAGE 44 Roots of Auxiliary Equation Repeated ·
Roots of the Auxiliary Equation Complex
Properties of the Symbolic Operator (D-α)
Particular Integral
Another Method of finding the Particular Integral ΙΟΙ
Variation of Parameters •
Method of Undetermined Coefficients •
tary Function
Cauchy's Linear Equation
Summary
CHAPTER VIII
LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 53 Change of Dependent Variable
Change of Independent Variable
Summary
CHAPTER IX
MISCELLANEOUS METHODS FOR SOLVING EQUATIONS OF HIGHER ORDER THAN THE FIRST 56 General Plan of Solution ·
Dependent Variable Absent
Independent Variable Absent
Linear Equations with Particular Integral Known •
Exact Equation Integrating Factor •
Transformation of Variables •
Summary ·
CHAPTER X
SYSTEMS OF SIMULTANEOUS EQUATIONS 63 General Method of Solution •
Systems of Linear Equations with Constant Coefficients
Systems of Equations of the First Order
Geometrical Interpretation •
Systems of Total Differential Equations •
Differential Equations of Higher Order than the First Reducible to Systems of Equations of the First Order
Summary •
INTEGRATION IN SERIES
Riccati's Equation ·
Gauss's Equation Hypergeometric Series ·
Primitives involving Arbitrary Functions •
CHAPTER XIII
Non-linear Partial Differential Equations of First Order Complete,
Special Methods
Summary
Special Method
89
SECTION PAGE
91
Non-homogeneous Linear Equations with Constant Coefficients
Condition that a Relation exist between Two Functions of Two Variables
INDEX
