Introduction to di▯erential equations (DEs)
1.1 What is a di▯erential equation?
A di▯erential equation or DE is an equation that contains derivatives. Typi-
cally, a DE provides a relationship between a function and its derivatives with respect
to one or more variables.
y (x) = 2(y(x)) + xy(x) (1.1)
@ u @ u @ u
+ + = u (1.2)
@x2 @[email protected]
d x dx 2t
dt2 + 3 dt+ 2x = e (1.3)
Dependent and Independent Variables:
Di▯erential equations have dependent and independent variables. In the case of
Equation 1.1, we call y = y(x) the dependent variable because the quantity y literally
depends on x. (If you want, think "y is a function of x" though the relationship
may not necessarily be that of a function.) Then, x is the independent variable.
Similarly, Equation 1.2 has an independent variable of u and independent variables 2
x and y. Equation 1.3 has an independent variable of x and an independent variable
of t. Sometimes, the relationship is explicitly written as in 1:1. Often though, it is
implied. To quickly determine which variable is independent, look at the variable
being di▯erentiated; those variables that are being di▯erentiated with respect to are
the independent variables.
Di▯erential equations are very useful for modelling real-world problems over a
wide range of disciplines. For example:
▯ Population dynamics in ecological systems (Biology)
▯ Quantities of reagents during a chemical reaction (Chemistry)
▯ Positions of masses interacting in a system over time (Physics)
Of course, Engineering applications can encompass any of these. For example, brie
consider the subject of mechanical resonance, a phenomenon that can occur when
a mechanical system oscillates (due to wind, etc) at a particular natural frequency
inherent to the system, a danger that can cause severe swaying or collapse of buildings,
bridges, or airplanes. The natural frequencies of a structure can often be determined
by using DEs. Thus, engineers might construct a model using DEs that avoid common
resonant frequencies that could lead to total collapse and then build based on that
In other words, DEs are important for all sorts of reasons beyond this course!
1.2 Describing Di▯erential Equations
There are many types of DEs. Eventually, we will be solving certain types of DEs,
but ▯rst it is important to understand how they may be classi▯ed. 3
Order of a DE:
The order of a DE is a number equal to the order of the highest derivative that
appears in the equation.
x (t) + 5x(t) = 0 (1.4)
is a third-order DE with dependent variable x and independent variable t.
Ordinary Di▯erential Equations (ODEs) :
These are di▯erential equations that contain only ordinary derivatives as opposed
to partial derivatives. ODEs generally contain a dependent variable and a single
v (x) ▯ v (x) = 0 (1.5)
is a fourth-order ODE with dependent variable v and independent variable x. Note
that the notation v(4)(x) could also be used to represent the fourth derivative of v with
respect to x here; more generally, v (n(x) is often used to denote the n thderivative of
v with respect to x.
Partial Di▯erential Equations (PDEs) :
These are di▯erential equations that contain partial derivatives. PDEs contain a
dependent variable and possibly many independent variables.
@ u @u
@x 2= @t (1.6)
is a PDE with dependent variable u and independent variables x and t.
Dealing with PDEs can be a much more di▯cult task than dealing with ODEs.
For most of this course, we will concern ourselves with ODEs. 4
In the general case, we will deal with ODEs of the form
y(n)= f(y (n▯1;y (n▯2);:::;y ;y;t); where y = y(t): (1.7)
We can always write ODEs in this form |with the highest-order derivative ap-
pearing by itself on one side |by dividing every term in the ODE by the coe▯cient
appearing in front of the highest-order derivative.
Linear ODEs :
These are ODEs in which the dependent variable appears only linearly. For an
equation in the form of 1.7, this means that the y, y , y , ::: never appear together in
a single term, nor are raised to any power except 1; they may only be multiplied by
constants or functions of the independent variable and added or subtracted from one
another. Equations including forms like sin(y) or cos(y ) are not linear. Note that
any forms of the independent variable may be found in a linear ODE! The
restrictions only lie on the dependent variable and its derivatives.
00 1 0
y + y + sin(t)y = 0 (1.8)
t ▯ 1
is a linear second-order ODE.
0 3 0 2
(x ) ▯ (x ) = tx (1.9)
is a non-linear ▯rst-order ODE, because there are derivatives of the dependent vari-
able raised to powers other than 1.
d y ▯ y dy = ex (1.10)