Basic Terminology

The study of differential equations is very broad, as it essentially describes any system which undergoes change. There are a few key words to know when discussing or describing these sorts of equations.

A differential equation is one that relates a function to its derivatives. For instance, `y=3y'-2` and `\left(y''\right)^2=\pi` are both differential equations.

An ordinary differential equation (ODE) is one that involves a function of one independent variable, while a partial differential equation (PDE) involves several independent variables. PDEs are generally far more difficult to solve.

The order of a differential equation is the highest order of derivation of the unknown function. For instance, `y^{(4)}-y''+y=0` is a fourth-order differential equation and `y'=y+x^4` is a first-order differential equation.

A differential equation is linear if and only if it can be written as `\sum_{i=0}^{n}a_i(x)y^{(n-i)}=f(x)` for a given order `n` and arbitrary functions `a_i(x)` and `f(x)`.
The equation `y'=3xy+2-x^3` is linear but `y^2=y''+3x` is not because of the `y^2` term.
Such an equation is homogeneous if and only if `f(x)=0`. For instance, the equation `y(x-1)=y''` is homogeneous, but `y'=x+y` is not.

An alternate use of the term homogeneous in the context of first-order differential equations is when describing those which can be written `\dfrac{\mathrm dy}{\mathrm dx}=F\left(\dfrac yx\right)` for some function `F`. This comes from the usage of the term "homogeneous" to describe general functions of two variables `f(x,y)` where `f(\lambda x,\lambda y)=\lambda^nf(x,y)` for some real "degree" `n`. ^[src]

The previous definition can be restated to say that a homogeneous first-order differential equation can be written `M(x,y)\,\mathrm dx + N(x,y)\,\mathrm dy=0` for any `M` and `N` that are homogeneous functions of the same degree. Note that rearranging gives `\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{M(x,y)}{N(x,y)}`, and the restriction on `M` and `N` gives `\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{M(\lambda x,\lambda y)}{N(\lambda x,\lambda y)}` for any `\lambda`. Thus, the right-hand side is unchanged when scaling `x` and `y` by the same amount, and this is only true for a function `F\left(\dfrac yx\right)` where the scaling is exactly cancelled-out by itself.

This general notion of invariance under multiplication of both `x` and `y` can be used to define homogeneity for higher-order differential equations, ^[src] and is the reasoning behind the use of the term in regards to linear ODEs (where `Cy` works as a solution to any equation solved by `y`).

TODO: Exact, separable, and autonomous differential equations