First-Order Differential Equations: Theory

First-Order Differential Equations: Theory, Methods and Applications provides comprehensive treatment of first-order ordinary differential equations for undergraduates and graduate students in mathematics, physics, engineering, and the applied sciences. The presentation balances theoretical foundations with practical solution techniques and scientific modeling.

Five chapters develop the existence and uniqueness theory for initial value problems including the Picard-Lindelِf theorem, Picard iteration, Peanos theorem, interval of definition, and Grِnwalls inequality. This framework provides essential preparation for advanced study in differential equations and dynamical systems.

The text systematically presents standard techniques for solving first-order equations: separable equations, linear equations, exact equations, homogeneous equations, and the method of integrating factors. Each chapter is developed through clear exposition, worked-out examples, and progressive exercises with answers.

Dedicated chapters demonstrate how differential equations arise from fundamental physical principles. Topics include population dynamics (exponential, logistic, Gompertz, threshold, Allee effect, and harvesting models), Newtons law of cooling, mixing problems and chemical kinetics, radioactive decay, motion under gravity with air resistance, and electrical circuit analysis.
A dedicated chapter on Eulers method introduces numerical approximation with error and stability analysis.