At its most broadest definition differential equations are equations that contain a derivative in some capacity. In application, these equations generally take the from:
We consider finding the function
The acronym ODE refers to ordinary differential equation. See the class notes index for a list of lectures and lecture notes.
It should be noted that differential equations are not a "clean" subject, there is no universal way to solve a differential equation and many are impossible to solve. Different approaches must be taken depending on the properties of the differential equation in question.
Much like the information lost when taking a derivative we cannot always pull all the necessary information about a solution from a differential equation.
Instead we break up our solutions into two main types: a general solution which does not provide a starting point for a line but instead essentially incudes all possible solutions and a particular solution which is a single solution picked from all the possibilities the general solution provides.
Slope fields are useful for showing general solutions.