Classification of differential equations
The order of a differential equation is the highest order of any differential contained in it.
Examples:
$\frac{\color{red}{\rm d}y}{\color{red}{\rm d}x}=ax$
is 1st order,
$\frac{{\rm d}^{\color{red}{3}}y}{{\rm d}x^3}+\frac{y}{x}=b$
is 3rd order, and
$\frac{\partial^{\color{red}{2}}z}{\partial x\partial y}+\frac{\partial z}{\partial x}+z=0$
is 2nd order.
An ordinary differential equation (ODE) contains differentials with respect to only one variable, partial differential equations (PDE) contain differentials with respect to several independent variables.
Examples:
$\frac{{\color{red}{\rm d}}y}{{\color{red}{\rm d}}x}=ax$
and
$\frac{{\color{red}{\rm d}}^3y}{{\color{red}{\rm d}}x^3}+\frac{y}{x}=b$
are ODE, but
$\frac{\color{red}{\partial}^2z}{\color{red}{\partial}x\color{red}{\partial}y}+\frac{\color{red}{\partial}z}{\color{red}{\partial}x}+z=0$
and
$\frac{\color{red}{\partial}z}{\color{red}{\partial}x}=\frac{\color{red}{\partial}z}{\color{red}{\partial}y}$
are PDE.
The straight and curly `d`s give it away if used properly. The real test is whether the dependent variable depends on just one or on more independent variables. In the examples above, we have y(x) but z(x,y).
Linear differential equations do not contain any higher powers of either the dependent variable (function) or any of its differentials, non-linear differential equations do.
Examples:
All of the examples above are linear, but
$\left(\frac{{\rm d}y}{{\rm d}x}\right)^{\color{red}{2}}=y$
isn't.
Note that
$\left(\frac{{\rm d}y}{{\rm d}x}\right)^2\neq\frac{{\rm d}^2y}{{\rm d}x^2}$
!
A differential equation is homogeneous if it contains no non-differential terms
and heterogeneous if it does.
Examples:
$\frac{{\rm d}y}{{\rm d}x}=\color{red}{ax}$
and
$\frac{{\rm d}^3y}{{\rm d}x^3}+\frac{{\rm d}y}{{\rm d}x}=\color{red}{b}$
are heterogeneous (unless the coefficients a and b are zero),
but
$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial y}$
is homogeneous.
A zero right-hand side is a sign of a tidied-up homogeneous differential equation, but beware of non-differential terms hidden on the left-hand side!
Solving heterogeneous differential equations usually involves finding a solution of the corresponding homogeneous equation as an intermediate step.