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IMA Journal of Applied Mathematics 1977 20(1):53-60; doi:10.1093/imamat/20.1.53
© 1977 by Institute of Mathematics and its Applications
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Remarks on the Linearization of Differential Equations

I. TOLSTOY {dagger}

School of Mathematics, The University Leeds

The Hamiltonian formalism and the theory of canonical transformations are used in this paper, first of all, to show that, given an ordinary non-linear differential equation it is always possible in principle to find a variable transformation reducing it to a linear equation, or a system of linear equations. The proof given is not to be construed as a general practical method for finding this transformation; it merely shows that such a transformation must always exist.

It is suggested that this may also hold true for partial differential equations. The conjecture is made plausible, in two cases, by the use of canonical transformation procedures for linearizing simple non-linear partial differential equations—one being a slight generalization of Burger's equation and the other an equation in three independent variables reminiscent of the Euler equations for fluid flow.


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