where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form
which is easy to solve by integration of the two members.
Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.
The term homogeneous was first applied to differential equations by Johann Bernoulli in section 9 of his 1726 article De integraionibus aequationum differentialium (On the integration of differential equations).
A first order differential equation of the form (a, b, c, e, f, g are all constants)
where af ≠ be
can be transformed into a homogeneous type by a linear transformation of both variables (α and β are constants):
Homogeneous linear differential equationsedit
A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or any derivative of it. A linear differential equation that fails this condition is called inhomogeneous.
Boyce, William E.; DiPrima, Richard C. (2012), Elementary differential equations and boundary value problems (10th ed.), Wiley, ISBN 978-0470458310. (This is a good introductory reference on differential equations.)
Ince, E. L. (1956), Ordinary differential equations, New York: Dover Publications, ISBN 0486603490. (This is a classic reference on ODEs, first published in 1926.)
Andrei D. Polyanin; Valentin F. Zaitsev (15 November 2017). Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems. CRC Press. ISBN 978-1-4665-6940-9.
Matthew R. Boelkins; Jack L. Goldberg; Merle C. Potter (5 November 2009). Differential Equations with Linear Algebra. Oxford University Press. pp. 274–. ISBN 978-0-19-973666-9.