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.

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.