Mejlbro L. Ordinary differential equations of first order

Bookboon.com, 2017. — 262 p.In the present volume we present the state-of-the-art of the theory of ordinary differential equations of first order as known in the middle of the twentieth century. The emphasis has been laid on solution formulæ, so the reader can get some idea of how to solve one of these classical equations, while topological methods have been more or less neglected. Some worked out examples illustrate the theory, and sometimes also the difficulties of the methods, in particular when we are forced to use the theorem of implicit given functions. Note, however, that if we use programs like MAPLE in such cases, it is often easy to sketch the solution curves by using a parametric description instead, while an explicit description may not be possible. In Chapter 1 we collect all the necessary mathematics, like the use of uniformly convergent sequences of functions, used in the iteration processes, Banach’s fixed point theorem, and how approximately to solve an implicitly given function in two variables, all needed in the following chapters. It should be mentioned that the Riccati equation has not been discussed in all details, because it is too closely connected with a class of ordinary differential equations of second order, which are outside the realm of this book. One may say that it belongs to the twilight zone between ordinary differential equations of first and second order.Some necessary auxiliary results.
First order differential equations and differential forms.
Separation of the variables.
The linear differential equation of first order.
Bernoulli’s equation.
Riccati’s equation.
The homogeneous case.
The homogeneous case where L(x, y) and M(x, y) are polynomials of degree 1.
Some simple implicit given differential equations.
Summary of solution methods and formulæ.