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But still more remarkable applications which Thomson made of his brother's integrating machine were to the mechanical integration of linear differential equations, with variable coefficients, to the integration of the general linear differential equation of any order,213 and, finally, to the integration of any differential equation of any order.

Every problem of mathematical physics which leads to a linear differential equation supplies an instance.

The special classes of discontinuous groups that have been dealt with Group of a linear differential equation. in the previous paragraphs arise directly from geometrical considerations.

A differential equation of this form is satisfied by the quotient of two independent integrals of the linear differential equation of the second order satisfied by the hypergeometric functions.

Conversely it should be noticed that any single linear differential equation d�x = u + vx + w dx , dt� dt where u, v, w are functions of t, by writing y for dx/dt, is equivalent with the two equations dx/dt = y, dy/dt = u + vx + wy.