We introduce how to classify various properties of differential equations.

There are many types of differential equations, and we classify them into different categories based on their properties. Let us quickly go over the most basic classification. We already saw the distinction between ordinary and partial differential equations:

Let us see some examples of ordinary differential equations:

And of partial differential equations:

If there are several equations working together, we have a so-called system of differential equations. For example,

is a simple system of ordinary differential equations. Maxwell’s equations for electromagnetics,

are a system of partial differential equations. The divergence operator and the curl operator can be written out in partial derivatives of the functions involved in the , , and variables.

In the first chapter, we will start attacking first order ordinary differential equations, that is, equations of the form . In general, lower order equations are easier to work with and have simpler behavior, which is why we start with them.

We also distinguish how the dependent variables appear in the equation (or system).

Another way to determine if a differential equation is linear is if the equation is a sum of terms, where each term is some function of the independent variables or some function of the independent variables multiplied by a dependent variable or its derivative. That is, an ordinary differential equation is linear if it can be put into the form

The functions , , …, are called the coefficients. The equation is allowed to depend arbitrarily on the independent variable. So is still a linear equation as and its derivatives only appear linearly. The equation is also linear, even though it is not initially in the correct form. From this equation, we can move the last term over to the right-hand side as a , and then it is in the correct form, with the term missing (or has coefficient zero).

All the equations and systems above as examples are linear. It may not be immediately obvious for Maxwell’s equations unless you write out the divergence and curl in terms of partial derivatives. Let us see some nonlinear equations. For example Burger’s equation,

is a nonlinear second order partial differential equation. It is nonlinear because and are multiplied together. The equation is a nonlinear first order differential equation as there is a second power of the dependent variable .

For example, the exponential growth equation, the wave equation, or the transport equation above are homogeneous. The mechanical vibrations equation above is nonhomogeneous as long as is not the zero function. Similarly, if the ambient temperature is nonzero, Newton’s law of cooling is nonhomogeneous. A homogeneous linear ODE can be put into the form

Compare to (classification:eqlingen) and notice there is no function .

If the coefficients of a linear equation are actually constant functions, then the equation is said to have constant coefficients. The coefficients are the functions multiplying the dependent variable(s) or one of its derivatives, not the function standing alone. A constant coefficient nonhomogeneous ODE is an equation of the form

where are all constants, but may depend on the independent variable . The mechanical vibrations equation above is a constant coefficient nonhomogeneous second order ODE. The same nomenclature applies to PDEs, so the transport equation, heat equation and wave equation are all examples of constant coefficient linear PDEs.

Finally, an equation (or system) is called autonomous if the equation does not explicitly depend on the independent variable. For autonomous ordinary differential equations, the independent variable is then thought of as time. Autonomous equation means an equation that does not change with time. For example, Newton’s law of cooling is autonomous, so is equation (classification:eqnonlinode). On the other hand, mechanical vibrations or (classification:eqlinex) are not autonomous.