Homogeneous differential equation

The term homogeneous differential equation has several distinct meanings.

One meaning is that a first-order ordinary differential equation is homogeneous (of degree 0) if it has the form

$\frac{dy}{dx} = F(x,y)$

where F(x,y) is a homogeneous function of degree zero; that is to say, that F(tx,ty) = F(x,y).

In a related, but distinct, usage, the term linear homogeneous differential equation is used to describe differential equations of the form

$Ly = 0 \,$

where the differential operator L is a linear operator, and y is the unknown function.

The remainder of this article is about homogeneous differential equations in the first sense defined above.

Solving homogeneous differential equations

By the definition above, it can be seen that F(tx,ty) = F(x,y) for all t, so t can be arbitrarily chosen to simplify the form of the equation. One CAN solve this equation by making a simple change of variables y = ux, and then using the product rule on the left hand side as follows,

$\frac{d(ux)}{dx} = x\frac{du}{dx} + u\frac{dx}{dx} = x\frac{du}{dx} + u$.

and then using the identity F(tx,ty) = F(x,y) to simplify the right hand side by choosing to set t to be 1 / x, transforming the original problem into the separable differential equation

$x\frac{du}{dx} + u = F(1,u)$

which can then be integrated by the usual methods.

See also

• Method of separation of variables

References

• Olver, P.J. (1995), Equivalence, invariants, and symmetry, Oxford University Press, ISBN 0521478111 , Example 6.20, pp. 188–189.

External links

• Homogeneous differential equations at MathWorld
• Homogeneous Differential equations
• Wikibooks: Differential Equations/First-Order/Substitution Methods