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Reducing a Differential Equation of a Special Form to a Homogeneous Equation

This Demonstration shows the reduction of a differential equation of the form to a homogeneous differential equation of the form . This case occurs if the system of linear equations , has a unique solution , ; then new variables are introduced by the equations , . If the system of linear equations has no solution or has infinitely many solutions, the differential equation reduces to an equation with separable variables.

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The equation is called homogeneous if and are homogeneous functions of of the same order. The equation can be reduced to the form . A function is called homogeneous of order if . An example: and are homogeneous of order 2, and is homogeneous of order 0.
The differential equation is not homogeneous in the usual sense of a linear differential equation having a right-hand side equal to zero, like .
References
[1] V. I. Smirnoff, Lectures in Higher Mathematics (in Russian), Vol. 2, Moscow: Nauka, 1967 pp. 19–21.
[2] L. E. Eljsgoljc, Differential Equations and Variational Calculus (in Russian), Moscow: Nauka 1969 pp. 26–27.
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