Defining Homogeneous Differential Equations

Question

I am putting together a list of types of first and second order differential equations and I am struggling with the definition of homogeneous and nonhomogeneous. Can anyone clarify the definitions for me please?

Answer 1

The word homogeneous, somewhat confusingly, is used in two different ways when describing differential equations.

In the case of first-order equations, a homogeneous equation is usually something of the form

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

i.e. an equation in which $y$ and $x$ appear only in the combination $\frac{y}{x}$. ‘Homogeneous’ in this case refers to the fact that if $y(x)$ is a solution then $\lambda y(\frac{x}{\lambda})$ is also a solution for any $\lambda \neq 0$.

An example of such an equation would be $$\frac{dy}{dx} = \frac{y^2 + 4xy - x^2}{4x^2}$$

and we can divide the numerator and the denominator by $x^2$ to obtain

$$\frac{dy}{dx} = \frac{(\frac{y}{x})^2 + 4(\frac{y}{x}) - 1}{4}$$

which meets our criteria for a first-order homogeneous equation.

In the case of second-order equations, it usually means something of the form

$$\frac{d^2y}{dx^2} + a(x) \frac{dy}{dx} + b(x)y = 0$$

The zero on the right hand side is what makes this a homogeneous equation. If $y_1$ and $y_2$ are solutions, then $A y_1(x) + B y_2 (x)$ is also a solution, for constants $A, B \in \mathbb{R}$.

A useful way of thinking about this is to observe that the left hand side defines a linear map on the vector space of differentiable functions, and so the solution space is the set of vectors mapped to the zero vector, i.e. the kernel of this linear map. The kernel is in particular a subspace and so is closed under addition and scalar multiplication (in this case by real numbers).

A inhomogeneous equation, then, is an equation where the right hand side is not zero. The general form is

$$\frac{d^2y}{dx^2} = + a(x) \frac{dy}{dx} + b(x)y = f(x)$$

These are harder to solve, and do not have the same properties as the homogeneous variant.