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### 0.3Classiﬁcation of diﬀerential equations

Note: less than 1 lecture or left as reading, §1.3 in [BD]

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:

• Ordinary diﬀerential equations or (ODE) are equations where the derivatives are taken with respect to only one variable. That is, there is only one independent variable.
• Partial diﬀerential equations or (PDE) are equations that depend on partial derivatives of several variables. That is, there are several independent variables.

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 governing 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.

The next bit of information is the order of the equation (or system). The order is simply the order of the largest derivative that appears. If the highest derivative that appears is the ﬁrst derivative, the equation is of ﬁrst order. If the highest derivative that appears is the second derivative, then the equation is of second order. For example, Newton’s law of cooling above is a ﬁrst order equation, while the Mechanical vibrations equation is a second order equation. The equation governing transversal vibrations in a beam,

is a fourth order partial diﬀerential equation. It is fourth order since at least one derivative is the fourth derivative. It does not matter that derivatives with respect to are only second order.

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). In particular, we say an equation is linear if the dependent variable (or variables) and their derivatives appear linearly, that is only as ﬁrst powers, they are not multiplied together, and no other functions of the dependent variables appear. In other words, 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. Otherwise the equation is called nonlinear. For example, an ordinary diﬀerential equation is linear if it can be put into the form

 (2)

The functions , , …, are called the coeﬃcients. The equation is allowed to depend arbitrarily on the independent variables. So

 (3)

is still a linear equation as and its derivatives only appear linearly.

All the equations and systems given 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

 (4)

is a nonlinear ﬁrst order diﬀerential equation as there is a power of the dependent variable .

A linear equation may further be called homogeneous, if all terms depend on the dependent variable. That is, if there is no term that is a function of the independent variables alone. Otherwise the equation is called nonhomogeneous or inhomogeneous. For example, Newton’s law of cooling, Transport equation, Wave equation, above are homogeneous, while Mechanical vibrations equation above is nonhomogeneous. A homogeneous linear ODE can be put into the form

Compare to (2) 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. That is, a constant coeﬃcient ODE is

where are all constants, but may depend on the independent variable . The Mechanical vibrations equation above is constant coeﬃcient nonhomogeneous second order ODE. 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 depend on the independent variable. Usually here we only consider ordinary diﬀerential equations and 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 (4). On the other hand, Mechanical vibrations or (3) are not autonomous.

#### 0.3.1Exercises

Exercise 0.3.1: Classify the following equations. Are they ODE or PDE? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant coeﬃcient? If it is an ODE, is it autonomous?
a)
b)
c)
d)
e)
f)

Exercise 0.3.2: If is a vector, we have the divergence and curl . Notice that curl of a vector is still a vector. Write out Maxwell’s equations in terms of partial derivatives and classify the system.

Exercise 0.3.3: Suppose is a linear function, that is, for constants and . What is the classiﬁcation of equations of the form .

Exercise 0.3.4: Write down an explicit example of a third order, linear, nonconstant coeﬃcient, nonautonomous, nonhomogeneous system of two ODE such that every derivative that could appear, does appear.

Exercise 0.3.101: Classify the following equations. Are they ODE or PDE? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant coeﬃcient? If it is an ODE, is it autonomous?
a)
b)
c)
d)
e)
f)

Exercise 0.3.102: Write down the general zeroth order linear ordinary diﬀerential equation. Write down the general solution.