PDE & Systems of ODE by Ravendra Kumar and Prof. B. R. Sharma is a well-structured textbook designed according to the CBCS syllabus for B.Sc./B.A. 3rd Semester Mathematics Honours of Indian universities. Published by Mahaveer Publications, the book offers a solid foundation in both Partial Differential Equations (PDEs) and Systems of Ordinary Differential Equations (ODEs), which form the core of applied mathematics and mathematical modelling.

The content covers essential methods for forming, solving, and interpreting differential equations. It introduces classical techniques such as separation of variables, linear and non-linear systems, homogeneous and non-homogeneous equations, eigenvalue methods, and standard PDE models used in physics, engineering, and real-world applications.

UNIT 1: Partial Differential Equations of First Order (PDE-I)

1.1 Definition of PDE

1.2 Order of a PDE

1.3 Degree of a PDE

1.4 Linear First-Order PDEs

1.5 Non-linear First-Order PDEs

1.6 Formation of Partial Differential Equations

Covers formation techniques, classification basics, and fundamental definitions.

UNIT 2: Lagrange’s Method & Quasilinear First-Order PDEs

2.1 Lagrange’s Equation

2.2 General Solutions of Lagrange’s Equation

2.3 Linear PDE in n Independent Variables

2.4 Integral Surfaces Passing Through a Given Point

2.5 Quasilinear First-Order PDEs

Focus on Lagrange’s method, characteristics, and geometric interpretation.

UNIT 3: Nonlinear First-Order PDEs—Standard Forms

3.1 Introduction

3.2 Standard Form I

3.3 Standard Form II

3.4 Standard Form III

3.5 Standard Form IV

Explores nonlinear PDEs and common reduction techniques.

UNIT 4: Charpit’s Method

4.1 Introduction

4.2 Charpit’s Auxiliary Equations

4.3 Jacobi’s Method

4.4 Working Method

4.5 PDEs with Four Independent Variables

Solution of nonlinear PDEs using Charpit–Jacobi frameworks.

UNIT 5: Classification of Second-Order PDEs & Canonical Form

5.1 Linear PDEs with Two Independent Variables

5.2 Classification of Second-Order PDEs (Elliptic, Parabolic, Hyperbolic)

5.3 Reduction to Canonical Form (Laplace Transformation Type)

5.4 Canonical Forms in Multiple Variables

This unit focuses on classification and transformation techniques.

UNIT 6: Method of Separation of Variables

6.1 Introduction

Fundamental technique used to solve PDEs under boundary value problems.

UNIT 7: Vibrating String Problem

7.1 One-Dimensional Wave Equation

7.2 Solution by Separation of Variables

7.3 Essential Series to Remember

Classical wave equation, eigenfunctions, and Fourier series.

UNIT 8: Heat, Wave & Laplace Equations

8.1 One-Dimensional Heat Equation

8.2 Heat Equation Solutions

8.3 One-Dimensional Wave Equation

8.4 Two-Dimensional Wave Equation

8.5 Laplace’s Equation

8.6 Laplace Equation in Polar Coordinates

8.7 Diffusion Equation

Covers major physical PDE models and their solutions.

UNIT 9: Systems of Ordinary Linear Differential Equations

9.1 Types of Linear Systems

9.2 Systems with Higher Derivatives

9.3 Normal Form for Systems

9.4 Differential Operator Techniques

9.5 Transforming nth Order ODE into Systems

9.6 Alternative Methods for Systems

9.7 Homogeneous & Non-homogeneous Linear Systems

9.8 Linear Combination of Solutions

9.9 Linear Dependence

9.10 Two-Variable Constant Coefficient Systems

A complete treatment of linear ODE systems, eigenvalue methods, and operator approaches.

UNIT 10: Numerical Methods for Differential Equations

10.1 Euler’s Method

10.2 Modified Euler’s Method

10.3 Taylor’s Method

10.4 Picard Successive Approximation

10.5 Runge–Kutta Method (General)

10.6 First-Order Runge–Kutta

10.7 Second-Order Runge–Kutta

10.8 Fourth-Order Runge–Kutta

10.9 RK Method for Simultaneous First-Order Systems

Provides numerical techniques essential for solving ODEs where analytical solutions are difficult.