A differential equation is linear if it is linear in the dependent variable and its derivatives: $y,y^\prime,y^{\prime\prime},\ldots$.

Example: $\sin(x)\frac{d^5y}{dx^5}+\frac{x}{x^4+2x+1}\frac{d^2y}{dx^2}+e^xy=\cos(x)$ is linear, although it is most definitely not linear in the independent variable x.

Example: $\left(\frac{dP}{dt}\right)^2=x$ is not linear since the derivative of the dependent variable P is squared.

A differential equation is separable if it is first order and can be written in the form $\frac{dy}{dx}=f(x)g(y)$. In other words, you must be able to write the derivative as a product, one of which does not contain the dependent variable and the other of which does not contain the independent variable.

Example: $\frac{dy}{dx}+xy=x$ is separable since we can write the equation in the form $\frac{dy}{dx}= x(1-y)$.

Example: $\frac{dz}{dt}+t^2z=t$ is not separable, rewriting we get $\frac{dz}{dt}=t(1-tz)$. Try as you might you will not be able to factor out the t from term $1-tz$.

Example: $(x^2+1)\frac{dy}{dx}-(x^2+1)y=(x^3+x)y$ is separable, if we divide through by $x^2+1:$ $\frac{dy}{dx}=y(x+1)$.

Example: $\frac{dy}{dx}=x$ is separable with $f(x) = x$ and $g(y)=1$.

A differential equation is autonomous if it is first order and can be written in the form $\frac{dy}{dx}=g(y)$.

Example: $\frac{dP}{dt}=P$ is autonomous with $g(P)=P$.

Example $\frac{dy}{dx}+x-\sin^2(y)x=\sin(y) + \cos^2(y)x$ is autonomous since $x-\sin^2(y)x=x(1-\sin^2(y))=x\cos^2(y)$, reducing the equation to $\frac{dy}{dx}=\sin(y)$

A linear differential equation is homogeneous if there are no terms that contain the independent variable alone.

Example: $\sin(x)\frac{d^5y}{dx^5}+\frac{x}{x^4+2x+1}\frac{d^2y}{dx^2}+e^xy=0$ is homogeneous.

Example: $\left(\frac{dy}{dt}\right)^2+t\frac{dy}{dt}=0$ is not homogeneous because it is not linear.

Example: $\frac{d^2y}{dt^2}+t\frac{dy}{dt}-t=0$ is not homogeneous because of the lone t.

Note that homogeneous is one of the most abused words in mathematics, there are multiple definitions for this word in use. This is the one we will use.