First Course In Differential Equations

A differential equation is an equation that involves the derivatives of a function as well as the function itself. If partial derivatives are involved, the equation is called a partial differential equation; if only ordinary derivatives are present, the equation is called an ordinary differential equation. Differential equations play an extremely important and useful role in applied math, engineering, and physics, and much mathematical and numerical machinery has been developed for the solution of differential equations. The order of the differential equation is the order of the highest derivative of the unknown function involved in the equation. The study of differential equations is a wide field in pure and applied mathematics, physics, meteorology, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods. A thorough, systematic first course in elementary differential equations for undergraduates in mathematics and science, requiring only basic calculus for a background, and including many exercises designed to develop students' technique in solving equations. Contents: Fractional Calculus; Fundamental Theorem of Calculus; Forcing Function; Analytic Element Method; Finite Element Method; Discrete Least Squares Meshless Method; Boundary Knot Method; Stochastic Differential Equation; Fast Multipole Method; Integral Transform; Lyapunov Stability; Homogeneous Functions; Differential Equations.