An Elementary Treatise on Differential Equations

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