Section 0.2 Classification of differential equations
Objectives
After this section, you will be able to:
-
Classify equation as ordinary or partial differential equations,
-
Identify whether an equation is linear or non-linear, and
-
Classify linear equations as homogeneous, non-homogeneous, or constant coefficient, as appropriate.
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:
Definition 0.2.1.
-
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:
\begin{align*}
\amp \frac{d y}{dt} = ky \amp \amp \text{(Exponential growth)} \\
\amp \frac{d y}{dt} = k(A-y) \amp \amp \text{(Newton's law of cooling)} \\
\amp m \frac{d^2 x}{dt^2} + c \frac{dx}{dt} + kx = f(t) \amp \amp \text{(Mechanical vibrations)}
\end{align*}
And of partial differential equations:
\begin{align*}
\amp \frac{\partial y}{\partial t} + c \frac{\partial y}{\partial x} = 0 \amp \amp \text{(Transport equation)} \\
\amp \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \amp \amp \text{(Heat equation)} \\
\amp \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} +
\frac{\partial^2 u}{\partial y^2} \amp \amp \text{(Wave equation in 2 dimensions)}
\end{align*}
If there are several equations working together, we have a so-called system of differential equations. For example,
\begin{equation*}
y' = x , \qquad x' = y
\end{equation*}
is a simple system of ordinary differential equations. Maxwell’s Equations for electromagnetics,
\begin{align*}
\nabla \cdot \vec{D} = \rho \amp \nabla \cdot \vec{B} = 0 \\
\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \amp \nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}
\end{align*}
are a system of partial differential equations. The divergence operator \(\nabla \cdot\) and the curl operator \(\nabla \times\) can be written out in partial derivatives of the functions involved in the \(x\text{,}\) \(y\text{,}\) and \(z\) variables.
In the first chapter, we will start attacking first order ordinary differential equations, that is, equations of the form
\(\frac{dy}{dx} = f(x,y)\text{.}\) 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).
Definition 0.2.2.
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. Otherwise, the equation is called
nonlinear.
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
\begin{equation}
a_n(x) \frac{d^n y}{dx^n} +
a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} +
\cdots
+
a_{1}(x) \frac{dy}{dx}
+
a_{0}(x) y = b(x) .\tag{0.2.1}
\end{equation}
The functions \(a_0\text{,}\) \(a_1\text{,}\) …, \(a_n\) are called the coefficients. The equation is allowed to depend arbitrarily on the independent variable. So
\begin{equation}
e^x \frac{d^2 y}{dx^2} +
\sin(x) \frac{d y}{dx} +
x^2 y
=
\frac{1}{x}\tag{0.2.2}
\end{equation}
is still a linear equation as \(y\) and its derivatives only appear linearly. The equation
\begin{equation*}
\cos(x) \frac{d^2y}{dx^2} - xy + \frac{e^x}{x} = 0
\end{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 \(-\frac{e^x}{x}\text{,}\) and then it is in the correct form, with the \(\frac{dy}{dx}\) 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,
\begin{equation*}
\frac{\partial y}{\partial t} +
y \frac{\partial y}{\partial x} =
\nu \frac{\partial^2 y}{\partial x^2} ,
\end{equation*}
is a nonlinear second order partial differential equation. It is nonlinear because \(y\) and \(\frac{\partial y}{\partial x}\) are multiplied together. The equation
\begin{equation}
\frac{dx}{dt} = x^2\tag{0.2.3}
\end{equation}
is a nonlinear first order differential equation as there is a second power of the dependent variable \(x\text{.}\)
Definition 0.2.3.
A linear equation may further be called
homogeneous if all terms depend on the dependent variable. That is, if no term is a function of the independent variables alone. Otherwise, the equation is called
nonhomogeneous or
inhomogeneous.
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 \(f(t)\) is not the zero function. Similarly, if the ambient temperature \(A\) is nonzero, Newton’s law of cooling is nonhomogeneous. A homogeneous linear ODE can be put into the form
\begin{equation*}
a_n(x) \frac{d^n y}{dx^n} +
a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} +
\cdots
+
a_{1}(x) \frac{dy}{dx}
+
a_{0}(x) y = 0 .
\end{equation*}
Compare to
(0.2.1) and notice there is no function
\(b(x)\text{.}\)
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 \(b(x)\) standing alone. A constant coefficient nonhomogeneous ODE is an equation of the form
\begin{equation*}
a_n \frac{d^n y}{dx^n} +
a_{n-1} \frac{d^{n-1} y}{dx^{n-1}} +
\cdots
+
a_{1} \frac{dy}{dx}
+
a_{0} y = b(x) ,
\end{equation*}
where \(a_0, a_1, \ldots, a_n\) are all constants, but \(b\) may depend on the independent variable \(x\text{.}\) 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
(0.2.3). On the other hand, mechanical vibrations or
(0.2.2) are not autonomous.
Exercises Exercises
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)
\(\displaystyle \sin(t) \frac{d^2 x}{dt^2} + \cos(t) x = t^2\)
Answer.
ODE, equation, second order, linear, non-homogeneous, not constant coefficient, not autonomous.
(b)
\(\displaystyle \frac{\partial u}{\partial x} + 3 \frac{\partial u}{\partial y} = xy\)
Answer.
PDE, equation, first order, linear, constant coefficient, non-homogeneous.
(c)
\(\displaystyle y''+3y+5x=0, \quad x''+x-y=0\)
Answer.
ODE, system, second order, linear, constant coefficient, homogeneous, autonomous.
(d)
\(\displaystyle \frac{\partial^2 u}{\partial t^2} + u\frac{\partial^2 u}{\partial s^2} = 0\)
Answer.
PDE, equation, second order, non-linear.
(e)
\(\displaystyle x''+tx^2=t\)
Answer.
ODE, equation, second order, non-linear, not autonomous.
(f)
\(\displaystyle \frac{d^4 x}{dt^4} = 0\)
Answer.
ODE, equation, fourth order, linear, constant coefficient, homogeneous, autonomous.
2.
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)
\(\displaystyle \frac{\partial^2 v}{\partial x^2} + 3 \frac{\partial^2
v}{\partial y^2} = \sin(x)\)
Answer.
PDE, equation, second order, linear, nonhomogeneous, constant coefficient.
(b)
\(\displaystyle \frac{d x}{dt} + \cos(t) x = t^2+t+1\)
Answer.
ODE, equation, first order, linear, nonhomogeneous, not constant coefficient, not autonomous.
(c)
\(\displaystyle \frac{d^7 F}{dx^7} = 3F(x)\)
Answer.
ODE, equation, seventh order, linear, homogeneous, constant coefficient, autonomous.
(d)
\(\displaystyle y''+8y'=1\)
Answer.
ODE, equation, second order, linear, nonhomogeneous, constant coefficient, autonomous.
(e)
\(\displaystyle x''+tyx'=0, \quad y''+txy = 0\)
Answer.
ODE, system, second order, nonlinear.
(f)
\(\displaystyle \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial s^2} + u^2\)
Answer.
PDE, equation, second order, nonlinear.
3.
If
\(\vec{u} = (u_1,u_2,u_3)\) is a vector, we have the divergence
\(\nabla \cdot \vec{u} =
\frac{\partial u_1}{\partial x} +
\frac{\partial u_2}{\partial y} +
\frac{\partial u_3}{\partial z}\) and curl
\(\nabla \times \vec{u} =
\Bigl(
\frac{\partial u_3}{\partial y} - \frac{\partial u_2}{\partial z} , ~
\frac{\partial u_1}{\partial z} - \frac{\partial u_3}{\partial x} , ~
\frac{\partial u_2}{\partial x} - \frac{\partial u_1}{\partial y} \Bigr)\text{.}\) Notice that curl of a vector is still a vector. Write out Maxwell’s equations in terms of partial derivatives and classify the system.
Answer.
Order 1, linear PDEs, with constant coefficients.
4.
Suppose
\(F\) is a linear function, that is,
\(F(x,y) = ax+by\) for constants
\(a\) and
\(b\text{.}\) What is the classification of equations of the form
\(F(y',y) = 0\text{.}\)
Answer.
First order, linear, homogeneous, constant coefficients.
5.
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.
Answer.
One example:
\(x''' + e^t x'' + tx' + x = t^2 + 1\) \(y''' + y'' + \sin(t)y' + y = e^{-t} - 2\)
6.
Write down the general
zeroth order linear ordinary differential equation. Write down the general solution.
Answer.
equation:
\(a(x) y = b(x)\text{,}\) solution:
\(y = \frac{b(x)}{a(x)}\text{.}\)
7.
For which
\(k\) is
\(\frac{dx}{dt}+x^k = t^{k+2}\) linear. Hint: there are two answers.
Answer.
\(k=0\) or
\(k=1\text{.}\)
8.
Write out an explicit example of a non-homogeneous fourth order, linear, constant coefficient differential equation. where all possible derivatives of the unknown function
\(y\) appear.
Answer.
One example:
\(y'''' + 8y''' + 3y'' + 2y' + 7y = e^{3t}\)
9.
Let \(x\text{,}\) \(y\text{,}\) and \(z\) be three functions of \(t\) defined by the system of differential equations
\begin{equation*}
x' = y \quad y' = z \quad z' = 3x - 2y + 5z + e^t
\end{equation*}
with initial conditions \(x(0) = 3\text{,}\) \(y(0) = -2\) and \(z(0) = 1\text{,}\) and let \(u(t)\) be the function defined by the solution to
\begin{equation*}
u''' - 5u'' + 2u' - 3u = e^t
\end{equation*}
with initial conditions \(u(0) = 3\text{,}\) \(u'(0) = -2\text{,}\) and \(u''(0) = 1\text{.}\)
(a)
Use the substitution
\(u=x\text{,}\) \(u' = y\text{,}\) and
\(u'' = z\) to verify that
\(x(t) = u(t)\) because they solve the same initial value problem.
(b)
What is the order of the system defining
\(x\text{,}\) \(y\text{,}\) and
\(z\) and how many components does it have?
Answer.
First order with three components.
(c)
What is the order of the equation defining
\(u\text{?}\) How many components does that have?
Answer.
Third order with one component.
(d)
How do these numbers relate to each other?
Answer.
The product is three in both cases. (
\(1 \times 3 = 3 \times 1\)).
