Symmetry Methods for Differential Equations

Description

mentioned on Math StackExchange here as more elementary than Bluman's books

contains hints and partial solutions to exercises

ch. 1 begins by quoting All's Well that Ends Well's Helena's "great floods have flown / From simple sources" (II.1)

also quotes G. K. Chesterton's The Higher Mathematics on p. 187:

There is a place apart
Beyond the solar ray,
Where parallel straight lines can meet
in an unofficial way.

and True Sympathy on p. 74:

Because I could not bear to make
An Algebraist cry
I gazed with interest at X
And never thought of Why.

This text, which is aimed at the advanced undergraduate level, is an introduction to group methods for differential equations. The book introduces the ideas and calculations involved without spending too much time on the pure mathematics which arises along the way. Throughout the text numerous examples are worked out in detail and the exercises have been well chosen. This is the most readable text on this material I have seen and I would recommend the book for self-study (as an introduction).
The first two-thirds of the book is concerned with scalar ordinary differential equations. Lie's method (using differential invariants) for integrating ordinary differential equations is covered in detail. This part of the book also introduces variational symmetries, contact symmetries, and some techniques of integration which use first integrals.
The last third of the book gives a brief introduction to symmetry methods for a scalar partial differential equation in two independent variables. Group invariant solutions (under a one-parameter subgroup) are introduced. The author shows how the action of the symmetry group on the space of solutions to the partial differential equation leads to the classification problem for invariant solutions. The procedure for solving the classification problem for solutions to a partial differential equation invariant under a one-parameter subgroup is given.
Infinitesimal methods are used exclusively throughout the book, except in the final chapter where the author introduces some of his work on discrete symmetries. In particular, it is shown how knowledge of infinitesimal symmetries can be used to simplify the search for discrete symmetries. Reviewed by Mark E. Fels