[1] 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.
[2] A first-order ordinary differential equation in the form: is a homogeneous type if both functions M(x, y) and N(x, y) are homogeneous functions of the same degree n.[3] That is, multiplying each variable by a parameter λ, we find Thus, In the quotient
, we can let t = 1/x to simplify this quotient to a function f of the single variable y/x: That is Introduce the change of variables y = ux; differentiate using the product rule: This transforms the original differential equation into the separable form or which can now be integrated directly: ln x equals the antiderivative of the right-hand side (see ordinary differential equation).
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): where For cases where af = be, introduce the change of variables u = ax + by or u = ex + fy; differentiation yields: or for each respective substitution.
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.