First-order ODE's: Terminology
Educational matters
Outcome
Define and/or apply the basic concepts pertaining to differential equations correctly.
What is a "differential equation"?
A differential equation (DE) is a mathematical equation with derivatives and functions.
Equal sign
Derivative
𝑑𝑦
= cos 3𝑥
𝑑𝑥
Let's have a closer look at this differential equation.
•
•
•
•
dy
dx
refers to the rate at which y changes with respect to x.
o dy
dx is NOT a fraction.
o 𝑑𝑦 and 𝑑𝑥 are differentials representing "small changes" in y and x
respectively.
x is the independent variable.
𝑦 = 𝑦(𝑥) is the dependent variable.
dy
= cos 3x predicts the slope of a tangent at a specific point.
The function
dx
We may classify differential equations in different ways. In this lesson we'll classify
differential equations:
•
•
•
•
By type,
By order or degree,
As linear or non-linear, and
As homogeneous or non-homogeneous.
Classification of differential equations
Ordinary vs. partial differential equations
An ordinary differential equation (ODE) contains ordinary derivatives with respect to one
independent variable.
First-order ODE's: Terminology – EL Voges
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Examples of ODE's
•
•
•
d 2 g
+ sin  = 0 .
dt 2 L
o The angular displacement  =  (t ) is a function of t only.
A
dh
= −k 0 2 gh .
Water drains from a tank according to Torricelli's law
dt
A
o The height h = h(t ) of the water in the tank is a function of t only.
The motion of a simple pendulum is governed by the DE
L
d 2q
dq q
+ R + = E (t ) is used to represent a basic LCR circuit.
2
dt
dt C
o The dependent variable q = q(t ) is a function of one variable (t) only.
A partial differential equation (PDE) contains partial derivatives with respect to two or
more independent variables.
Examples of PDE's
•
•
☂
  2u  2u 
u
=  2 + 2 .
t
 x y 
o The heat distribution u = u( x; y; t ) is a function if time and the space variables
x and y.
2
 B 1 2 B
=
is used to illustrate the propagation of the magnetic component B of an
x 2 c 2 t 2
electromagnetic wave in a medium.
o The dependent variable B = B( x; t ) is a function of x and t.
The distribution of heat in a plate is represented by
We'll investigate the derivation and solution of these differential equations, and more,
in detail later in this course.
Degree vs. order
•
•
The order of a differential equation refers to the highest derivative in the equation.
The degree of a differential equation refers to the power of the highest derivative in
the equation.
Consider the following ODE. It is a second-order ODE of degree one.
Derivative raised to the power 4
⇒ NOT highest-order derivative
⇒ NOT fourth degree ODE
Second order derivative
⇒ second order ODE
4
Highest-order derivative
to power 1 ⇒ First degree
order ODE
d 2q
 dq  q
+ 3   + = 25
2
dt
 dt  5
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Examples illustrating the difference between the order and the degree of a DE
•
( y ''')
3
− 5 x ( y ') = e x + 1 :
4
Third order ODE of degree 3
3
d4y
 dy 
+ 3t 2   − ( sin t ) y = 0 : Fourth order ODE of degree 1
4
dt
 dt 
•
5t
•
 dy 
3t   − ( sin t ) y 6 = 0 : First order ODE of degree 3
 dt 
3
2
Classification as linear or non-linear
A differential equation is linear if it has the following form:
dny
d n −1 y
dy
an ( x ) n + an −1 ( x ) n −1 + ... + a1 ( x ) + a0 ( x ) y = g ( x), an  0
dx
dx
dx
Note the dependent variable and its derivatives are linear (power 1). An equation which is not
linear is called non-linear.
Examples of linear/non-linear DE's
•
dy
+ 2 y = 5 ( x + 1)
dx
o A first-order linear ODE because
dy
dx
and y are linear (to the power 1), their
coefficients are constants, and there is no y (the dependent variable) on the
right.
•
3u ' ( t ) − u sin t = 0
o A first-order linear ODE because u '(t ) = du dt and u are to the power 1, their
coefficients are a constant and a function of t only, and there is no u on the
right-hand side.
2
•
d3y
 dy 
+ 4   − y = ex
3
dx
 dx 
o Non-linear because of ( dy dx ) .
2
•
u '''(t ) − 3u = eu sin t
o Non-linear because there is a function of the dependent variable u on the RHS.
o It may be written as u '''(t ) − 3u − eu sin t = 0 , but because of eu the DE is still
non-linear.
Classification as homogeneous or non-homogeneous
In English the word "homogeneous" means "of the same kind" or "alike". In the context of
differential equations, we distinguish between two different meanings.
First-order ODE's: Terminology – EL Voges
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1.
A TYPE OF FIRST -ORDER DE
A homogeneous function is a function such that f ( zx; zy) = z n f ( x; y) where z is a real
constant. The function is homogeneous of degree n.
•
f ( x; y) = x2 + xy
o
•
•
f ( zx; zy ) = ( zx) 2 + ( zx)( zy ) = z 2 ( x 2 + xy ) = z 2 f ( x; y )
o Thus, f ( x; y) is homogeneous of degree 2.
g ( x; t ) = x + t
o g ( zx; zt ) = ( zx) + ( zt ) = z ( x + t ) = zg ( x; t )
o  g ( x; t ) is homogeneous of degree 1.
p( x; y) = 3x + y 2
o p( zx; zy) = 3( zx) + ( zy)2 = z(3x + zy 2 )  zp( x; y)
o Thus, p( x; y) is non-homogeneous.
A first-order DE is algebraically homogeneous in x and y if
dy M ( x; y )
=
dx N ( x; y )
where M ( x; y) and N ( x; y) are homogeneous functions of the same degree.
•
•
☂
2.
dy x + y
=
is an algebraically homogeneous ODE in x and y.
dx x − y
o Both the numerator and denominator are homogeneous functions of degree 1.
dy
xy
=
is non-homogeneous in x and y.
dx x + y
o The numerator is a homogeneous function of degree 2, and the denominator is
of degree 1.
We sometimes leave out the "algebraically" and refer to these ODE's as homogeneous. Don't
confuse the "(algebraically) homogenous ODE" here with the "homogeneous ODE" in the next
section.
THE RIGHT-HAND SIDE OF A DE IN STANDARD FORM
The second interpretation refers to the right-hand side of a DE in standard form.
In the standard form: Arrange all the terms of a linear DE containing the dependent variable
on the left of the equal sign and the terms without the dependent variable on the right. In
general, a second-order ODE will have the form
a
d2y
dy
+ b − y = f (t ); a  0
2
dt
dt
with a, b and c are functions of t or constants, and 𝑓(𝑡) a function of t.
If 𝑓(𝑡) ≠ 0, DE is said to be non-homogeneous DE’s. When 𝑓(𝑡) = 0, the DE is
homogeneous.
First-order ODE's: Terminology – EL Voges
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Examples of homogeneous/non-homogeneous DE's
The ODE
d 2i
di
− 5 + 6i = 0
2
dt
dt
governs the current 𝑖 = 𝑖(𝑡) in an imaginary circuit. This is a homogeneous ODE because,
when written in standard form, the right-hand side is zero.
However,
d 2i
di
− 5 + 6i = 1
2
dt
dt
is non-homogeneous; note the 1 on the right-hand side. Although the left-hand sides of the
two second-order ODE's are the same, the solutions are different:
i = 3e2t − 2e3t
for the homogeneous DE above and
5
5
1
i = e 2 t − e 3t +
2
3
6
will satisfy the non-homogeneous DE above.
☛
You must be able to prove/verify those are indeed solutions of the DE's.
To determine the solutions above I used something called initial conditions.
Initial conditions vs. boundary conditions
Condition
A governing differential equation represents a phenomenon in a specific region. For example,
the heat equation may describe the distribution of heat in a plate.
A condition provides additional information we may use to determine the particular solution
of a differential equation, that is, to determine the actual value(s) of the arbitrary constant(s)
in the general solution.
Types of conditions
In this course we'll distinguish between boundary and initial conditions. Let y = y( x) .
•
Boundary condition
o Conditions that must be satisfied at the boundaries of the region governed by
the differential equation.
o 𝑦(1) = 3 and 𝑦(4) = −1 are two examples of boundary conditions.
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•
☝
o The value of y is given for different values of x.
Initial conditions
o Conditions that must be satisfied at specific points.
o 𝑦(0) = 3 and 𝑦 ′ (0) = −2 are two examples of initial conditions.
o The value of y is given for the same value of x.
o "Initial" usually refers to "at the beginning".
▪ When time is the independent variable, this condition usually, but not
always, implies 𝑡 = 0.
When you have only one condition, it is quite often difficult to decide whether it is a
BC or an IC. The context may assist with the classification. See the wave equation
below.
In this module we differentiate between
•
•
Boundary value problems (BVP's)
o A differential equation together with specified boundary conditions
Initial value problems (IVP's)
o A differential equation with specified initial conditions
Remember …
•
•
Quite often, as with the one-dimensional wave equation, we have a mixture of
boundary and initial conditions.
For every order of derivative, we need one condition. The wave equation
 2u
1  2u
=
−
has two second-order derivatives, and hence we usually need four
x 2
c 2 t 2
conditions to determine the particular solution of the wave equation.
An example of boundary and initial conditions
Keep the following in mind.
➢ Assume 𝑥 = 𝑥(𝑡) represents displacement. Then
𝑑𝑥
𝑑2 𝑥
o 𝑥̇ = represents velocity and 𝑥̈ = 2 represents
𝑑𝑡
𝑑𝑡
acceleration.
➢ Let 𝑢 = 𝑢(𝑥; 𝑡) represent displacement in space. Then
𝜕𝑢
o 𝑢𝑡 = 𝜕𝑡 represents the velocity at the point x.
Now consider the stretched string of length 80 with fixed endpoints
shown Figure 1.
Boundaries
Figure 1 The string with fixed ends
When the string is set in motion, the motion is governed by the one-dimensional wave
equation, a second-order partial differential equation,
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 2u 1  2u
=
x 2 64 t 2
subject to the boundary conditions
u(0; t ) = 0 t  0
Displacement at 𝑥 = 0 (left-hand side) is 0 for all t
u(80; t ) = 0 t  0
Displacement at 𝑥 = 80 (right-hand side) is 0 for all t
and initial conditions
 3x
0  x  40
 80 ,
u ( x;0) = 
3 − 3x , 40  x  80
 80
ut ( x;0) = 0 x [0;80] .
Initial displacement is a piecewise-defined function as
shown in Figure 2
Initial velocity is 0, that is, the string is released from
rest
Figure 2 The initial displacement
☂
We'll return to the solution of this partial differential equation later in this course.
The solution of a differential equation
What is a solution of a DE?
An algebraic equation has numbers as solutions. For example, x2 − 2 x + 3 = 0 has two
complex numbers as solution: x = 1  2 j .
A solution of a differential equation is a relation that satisfied the DE.
•
☛
The solution may be a function in some instances.
Revision: A function is a special type of relation. When is a relation a function?
General solution vs. particular solution
Consider the basic first-order ODE
dy
= cos 2 x
dx
From previous math courses you know how to solve for y: use integration.
y =  cos 2 xdx
1
= sin 2 x + c
2
With additional information, say y(0) = 3 , we may calculate the value of c
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1
3 = sin 0 + c
2
c = 3
1
 y = sin 2 x + 3
2
•
•
•
General solution
1
o The solution y = sin 2 x + c because of the arbitrary constant c.
2
Particular solution
1
o y = sin 2 x + 3 because there are no arbitrary constants in the solution; we
2
used additional information, called a condition, to determine the value of the
constant.
Trivial solution
o The solution 𝑦(𝑥) = 0 of a DE, that is, the solution is zero for all values of x.
Explicit vs. implicit
The solution of a DE may be given in implicit or explicit form.
•
•
Explicit form
o The dependent variable is expressed in terms of the independent variable, that
is, y = f ( x)
Implicit form
o It is not possible to change the subject of the solution and the solution has the
general form f ( x; y) = 0
An example of two formats
The solution of
dy
dx
= − x y may be written as:
•
x2 + y 2 = c (explicit form)
•
y =  c − x 2 (implicit form)
Types of solutions
We'll investigate three formats of solutions in this study unit.
Exact/analytical solution
•
•
•
The solution obtained using an analytical method such as separation of variables.
The solution is a relation, which may also be a function.
The solution is a continuous function or has a finite number of discontinuities.
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Graphical solution
A graph is used to assist with the interpretation of a solution.
Examples of the graphs of solutions
The four graphs below depict four different physical phenomena. Can YOU name them?
We'll expect you to do so.
y
3
2
1
0
1
2
3
4
5
6
x
-1
-2
x(t)
t
Numerical solution
Use a numerical technique such as one of the Runge-Kutta methods.
•
•
The solution is an approximation only.
The solution is given for discrete values of x only.
Example of a numerical solution
Using a numerical method to solve
dy
= x 2 + y 2 subject to y(0) = 1 we may obtain the
dx
solution
xn
0.0
0.2
0.4
0.6
0.8
1.0
☀
yn
1
1.2
1.496
1.9756032
2.828204801
4.55595328
The analytical solution of this first-order ODE with initial condition contains so-called
Bessel funtions and we may re-visit it at a later stage.
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☛
We'll return to numerical solutions later in this module.
Mathematical models
A mathematical model is the representation of a "word sum" in terms of mathematical
formulas. In this module the "translation" of English into Mathematics will probably result in
a model consisting of a differential equation and enough boundary and/or initial conditions to
determine the particular solution.
The steps:
1.
2.
3.
4.
Make assumptions about, for example, the forces acting on the body.
Set up the model, that is, the formulas.
Solve the model.
Analyse/interpret the solution.
We'll do a lot of modelling, i.e. applications, in this module.
Next …
Solve models governed by ODE's using direct integration.
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