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Concept
Homogeneous differential equation
Summary
A differential equation can be homogeneous in either of two respects.
A first order differential equation is said to be homogeneous if it may be written
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 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. 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):
Linear differential equation
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.
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