Ordinary and Partial Differential Equations

PART I: ELEMENTARY DIFFERENTIAL EQUATIONS

1. Differential Equations: Their Formation and Solutions
2. Equations of First Order and First Degree
3. Trajectories
4. Equations of the First Order but Not of the First Degree and Singular Solutions and Extraneous Loci
5. Linear Differential Equations with Constant Coefficients
6. Homogeneous Linear Equations or Cauchy-Euler Equations
7. Method of Variation of Parameters
8. Ordinary Simultaneous Differential Equations
9. Exact Differential Equations and Equations of Special Forms
10. Linear Equations of Second Order
11. Applications of Differential Equations
12. Miscellaneous Methods and Existence and Uniqueness Theorem for Solutions of First Order Initial Value Problems
PART II: ADVANCED ORDINARY DIFFERENTIAL EQUATIONS, FOURIER SERIES AND SPECIAL FUNCTIONS

1. Picard's Iterative Method, Picard's Theorem and Existence and Uniqueness of Solutions to First Order Initial Value Problems
2. Simultaneous Equations of the Form (dx)/P =(dy)/Q =(dz)/R
3. Total (or Pfaffian) Differential Equations
4. Beta and Gamma Functions
5. Chebyshev Polynomials
6. Fourier Series
7. Power Series
8. Integration in Series
9. Legendre Polynomials
10. Legendre Functions of the Second Kind—Qn(x)
11. Bessel Functions
12. Orthogonal Sets of Functions and Strum Liouville Problem

PART III: PARTIAL DIFFERENTIAL EQUATIONS
1. Origin of Partial Differential Equations
2. Linear Partial Differential Equations of Order One
3. Non-linear Partial Differential Equations of Order One
4. Homogeneous Linear Partial Differential Equations with Constant Coefficients
5. Non-homogeneous Linear Partial Differential Equations with Constant Coefficients
6. Partial Differential Equations Reducible to Equations with Constant Coefficients
7. Partial Differential Equations of Order Two with Variable Coefficients
8. Classification of P.D.E. Reduction to Canonical or Normal Forms Riemann Method
9. Monge's Methods
10. Transport Equation
11. Cauchy Initial Value Problem for Linear First Order Partial Differential Equations
Miscellaneous problems based on Part III of the book

• A new chapter on "Miscellaneous Methods and Existence and Uniqueness Theorem for Solutions of First Order Initial Value Problems
• Clear exposition of Picard's theorem and Picard's iterative method of successive approximations
• Detailed discussion on Lipschitz condition, Lipschitz constant, Lipschitz continuous function, Gronwall inequality and existence and uniqueness of solutions to first order initial value problems
• For practice a number of exercises including questions asked in different university examinations, GATE, CSIR-UGC (NET) and other competitive examinations