#### Overview of this book

Python, one of the world's most popular programming languages, has a number of powerful packages to help you tackle complex mathematical problems in a simple and efficient way. These core capabilities help programmers pave the way for building exciting applications in various domains, such as machine learning and data science, using knowledge in the computational mathematics domain. The book teaches you how to solve problems faced in a wide variety of mathematical fields, including calculus, probability, statistics and data science, graph theory, optimization, and geometry. You'll start by developing core skills and learning about packages covered in Python’s scientific stack, including NumPy, SciPy, and Matplotlib. As you advance, you'll get to grips with more advanced topics of calculus, probability, and networks (graph theory). After you gain a solid understanding of these topics, you'll discover Python's applications in data science and statistics, forecasting, geometry, and optimization. The final chapters will take you through a collection of miscellaneous problems, including working with specific data formats and accelerating code. By the end of this book, you'll have an arsenal of practical coding solutions that can be used and modified to solve a wide range of practical problems in computational mathematics and data science.
Preface
Basic Packages, Functions, and Concepts
Free Chapter
Mathematical Plotting with Matplotlib
Working with Randomness and Probability
Geometric Problems
Finding Optimal Solutions
Miscellaneous Topics
Other Books You May Enjoy

# Solving systems of differential equations

Differential equations sometimes occur in systems consisting of two or more interlinked differential equations. A classical example is a simple model of the populations of competing species. This is a simple model of competing species labeled P (the prey)and W (the predators) given by the following equations:

The first equation dictates the growth of the prey species P, which, without any predators, would be exponential growth. The second equation dictates the growth of the predator species W, which, without any prey, would be exponential decay. Of course, these two equations are coupled; each population change depends on both populations. The predators consume the prey at a rate proportional to the product of their two populations, and the predators grow at a rate proportional to the relative abundance of prey (again the product of the two populations).

In this recipe, we will will analyze...