Why We Wrote This Book
Like indefinite integrals, if the occasional differential equation can be solved in terms of familiar functions, we should solve it. Thus, sections of this book are devoted to the analytical solutions of differential equations. Like indefinite integrals, most differential equations cannot be solved in terms of familiar functions. When this occurs, we try to understand the solution by qualitative means (monotonicity, concavity, symmetry, singularities, existence and uniqueness of solutions) and by quantitative means (numerical solutions). How do we know the information we obtain about a differential equation is correct? The simple answer is to use everything we have at our disposal—analytical solutions, quantitative calculations, and qualitative arguments. We treat differential equations from a global point of view, subject to many different mathematical procedures.
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