This book is a dive into the world of partial differential equations (PDEs), differential equations that involve several variables and their partial derivatives. The text focuses on generalizing techniques for solving ordinary differential equations to PDEs. PDEs are notoriously difficult to solve and are used to describe many physical phenomena such as vibrating membranes and fluid dynamics. Because of this, their applications are widespread across a range of scientific disciplines. The author begins with the basics of PDEs, which provides an accessible entry point for those less familiar with the subject. The text includes chapters covering linear and nonlinear equations, existence theorems, variation of parameters, equations of mathematical physics, contact transformations, and more. The author employs a combination of theoretical explanations, examples, and exercises to help readers develop a thorough understanding of PDEs. Whether you're a student looking to master the fundamentals or a researcher seeking to expand your knowledge, this book offers a deep exploration of the subject.

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Excerpt from Theory of Differential Equations, Vol. 5: Partial Differential Equations

One chapter is devoted to the generalisation of integrals which involve some arbitrary parameters, and another to the discussion of characteristics of equations of the second order. The investigations in both of these chapters are clearly incomplete: they could be continued along lines that lead to the complete classification of integrals of equations of the first order.

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