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This chapter describes a simple scheme for the analysis of mathematical equations which relies on using invariants and makes it possible to simplify algebraic equations, reduce the order of ordinary differential equations (or integrate them), and find exact solutions of nonlinear partial differential equations. Invariants are constructed by searching for transformations that preserve the form of the equations; the notions and complex techniques of symmetry analysis (see Chapter 9) are not used here. Numerous examples of solving specific differential equations are given. It is significant that even with the simplest linear transformations of translation and scaling, as well as their compositions, the number of solvable ordinary differential equations (or those admitting order reduction) that can be described in a unified way is more than those discussed in the overwhelming majority of available textbooks. For nonlinear equations of mathematical physics, this approach makes it possible to find all of the most common invariant solutions. To use this simple method, one does not have to have a strong mathematical background?what is required is to be able to solve simple algebraic equations (and system of equations) and differentiate. To distinguish it from the classical group analysis method, the approach presented in this chapter will be called the method of invariants.
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