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:
\begin{equation*}
\begin{aligned}
& \frac{d y}{dt} = ky , & & \text{(Exponential growth)} \\
& \frac{d y}{dt} = k(A-y) , & & \text{(Newton's law of cooling)} \\
& m \frac{d^2 x}{dt^2} + c \frac{dx}{dt} + kx = f(t) . & &
\text{(Mechanical vibrations)}
\end{aligned}
\end{equation*}
And of partial differential equations:
\begin{equation*}
\begin{aligned}
& \frac{\partial y}{\partial t} + c \frac{\partial y}{\partial x} = 0 , & &
\text{(Transport equation)} \\
& \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} , & &
\text{(Heat equation)} \\
& \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} +
\frac{\partial^2 u}{\partial y^2} . & &
\text{(Wave equation in 2 dimensions)}
\end{aligned}
\end{equation*}
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 system of ordinary differential equations. Maxwell’s equations for electromagnetics,
\begin{equation*}
\begin{aligned}
& \nabla \cdot \vec{D} = \rho, & & \nabla \cdot \vec{B} = 0 , \\
& \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}, &
& \nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t} ,
\end{aligned}
\end{equation*}
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.
The next bit of information is the order of the equation (or system). The order is 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,
\begin{equation*}
a^4 \frac{\partial^4 y}{\partial x^4} + \frac{\partial^2 y}{\partial t^2} = 0,
\end{equation*}
is a fourth order partial differential equation. It is fourth order as at least one derivative is the fourth derivative. It does not matter that the derivative in \(t\) is only of second order.
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). In particular, 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. The equation is a sum of terms, where each term is a function of the independent variables or a function of the independent variables multiplied by a dependent variable or its derivative. Otherwise, the equation is called nonlinear. 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{2}
\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{3}
\end{equation}
is a linear equation as \(y\) and its derivatives only appear linearly.
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{4}
\end{equation}
is a nonlinear first order differential equation as there is a second power of the dependent variable \(x\text{.}\) Another nonlinear ODE is the pendulum equation
\begin{equation}
\theta''+\sin(\theta)=0 ,\tag{5}
\end{equation}
which is nonlinear as the dependent variable \(\theta\) appears inside a \(\sin\) function. Nonlinear equations are notoriously difficult to solve and their solutions may behave in strange an unexpected ways. Perhaps you have heard of chaos theory and the butterflies in the Amazon causing hurricanes in the Atlantic, all due to nonlinear equations. So sometimes we study similar linear equations, such as \(\theta''+\theta=0\text{,}\) instead.
A linear equation is further 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
(2) 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 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 are the equations
(4) and
(5). On the other hand, mechanical vibrations (as long as
\(f(t)\) is nonconstant) or
(3) are not autonomous. A general first order autonomous ODE would have the form
\begin{equation*}
\frac{dx}{dt} = f(x) .
\end{equation*}
Subsection 0.3.1 Exercises
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?
\(\displaystyle \displaystyle \sin(t) \frac{d^2 x}{dt^2} + \cos(t) x = t^2\)
\(\displaystyle \displaystyle \frac{\partial u}{\partial x} + 3 \frac{\partial u}{\partial y} = xy\)
\(\displaystyle \displaystyle y''+3y+5x=0, \quad x''+x-y=0\)
\(\displaystyle \displaystyle \frac{\partial^2 u}{\partial t^2} + u\frac{\partial^2 u}{\partial s^2} =
0\)
\(\displaystyle \displaystyle x''+tx^2=t\)
\(\displaystyle \displaystyle \frac{d^4 x}{dt^4} = 0\)
Exercise 0.3.2.
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.
Exercise 0.3.3.
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 \(F(y',y) = 0\text{.}\)
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?
\(\displaystyle \displaystyle \frac{\partial^2 v}{\partial x^2} + 3 \frac{\partial^2
v}{\partial y^2} = \sin(x)\)
\(\displaystyle \displaystyle \frac{d x}{dt} + \cos(t) x = t^2+t+1\)
\(\displaystyle \displaystyle \frac{d^7 F}{dx^7} = 3F(x)\)
\(\displaystyle \displaystyle y''+8y'=1\)
\(\displaystyle \displaystyle x''+tyx'=0, \quad y''+txy = 0\)
\(\displaystyle \displaystyle \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial s^2} + u^2\)
Answer.a) PDE, equation, second order, linear, nonhomogeneous, constant coefficient.
b) ODE, equation, first order, linear, nonhomogeneous, not constant coefficient, not autonomous.
c) ODE, equation, seventh order, linear, homogeneous, constant coefficient, autonomous.
d) ODE, equation, second order, linear, nonhomogeneous, constant coefficient, autonomous.
e) ODE, system, second order, nonlinear.
f) PDE, equation, second order, nonlinear.
Exercise 0.3.102.
Write down the general zeroth order linear ordinary differential equation. Write down the general solution.
Answer.equation: \(a(x) y = b(x)\text{,}\) solution: \(y = \frac{b(x)}{a(x)}\text{.}\)
Exercise 0.3.103.
For which \(k\) is \(\frac{dx}{dt}+x^k = t^{k+2}\) linear. Hint: there are two answers.
