Understanding Differential Equations

Here, the emphasis is on understanding differential equations. Not all differential equations can be solved by applying the standard techniques. Here too the numerical methods and geometrical insights make the student appreciate what it means to solve a general differential equation—for this purpose, a geometrical description is included, and Python codes are provided. Simple examples illustrate how geometrical insights aid in arriving at a solution. Exercises are given that address problems which do not yield to standard methods. A basic problem I have found is this: when introduced to calculus, the student learns that the usual Riemann integral is a limit of a sum—the Darboux sum. But soon, readymade formulae are invoked, and the student becomes adept at applying standard techniques—like integration by parts, substitution, etc.—and forgets that the integral is a sum. Although these techniques are useful, the approach has limited utility. This book serves to remind the student through a numerical approach that an integral is a sum. Non-standard examples are provided which illustrate this point. This chapter also covers standard equations that occur in physics, first and second order differential equations, and linear partial differential equations.