Homogeneous Meaning in Homogeneous Differential Equations
Description
$$ a_{n}(x)\dfrac{d^ny}{dx^n}+a_{n-1}(x)\dfrac{d^{n-1}y}{dx^{n-1}}+ \cdots + a_{1}(x)\dfrac{dy}{dx}+a_{0}(x)y=f(x) $$
When a differential equation is as above, if $f(x)=0$, it is called homogeneous $f(x) \ne 0$ if not, it is called non-homogeneous or inhomogeneous. Consider the following simple example of a 2nd order linear differential equation.
$$ ay^{\prime \prime}+by^\prime +cy=g(t) $$
Here, if $g(t)$ equals $0$, it is homogeneous; if $0$ is not met, it is non-homogeneous. To elaborate on the term homogeneous, it implies that the orders are the same. A homogeneous equation means an equation with the same orders. Here, having the same orders means that the orders of the dependent variable and its derivatives are all the same as understood. Let’s look at the formula below.
$$ a\left( y^{\prime \prime} \right)^{\color{blue}1}+b\left( y^\prime \right)^{\color{blue}1} + c\left( y \right)^{\color{blue}1} =0 =0 \left( y \right)^{\color{red}1} $$
$$ a\left( y^{\prime \prime} \right)^{\color{blue}1}+b\left( y^\prime \right)^{\color{blue}1} + c\left( y \right)^{\color{blue}1} =g(t) =g(t) \left( y \right)^{\color{red}0} $$
Looking at the first equation, when $g(t)=0$, all terms can be expressed as an equation where the order of the dependent variable and its derivatives are $1$. That is, all terms become the same order. Therefore, it is called homogeneous. Looking at the second formula, because of $g(t)$ which is not $0$, only the term $g(t)$ is of order $0$. That is, one term of a different order is created. Thus, it is called non-homogeneous.