Ordinary Differential Equations
Lecture 1: Introduction and direction fields. Basic
concepts of ODEs. Initial value problems and
boundary value problems.
Aishabibi Dukenbayeva
a.dukenbayeva@sdu.edu.kz
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
1 / 33Ba
Course: MAT 205 Ordinary differential equations
Instructor: Aishabibi Dukenbayeva
email: a.dukenbayeva@sdu.edu.kz
Assessments:
Midterm 1: 7th week of study 20p
Midterm 2: 13th week of study 20p
Homework: 20p
Final: 40p
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Course description
In this introductory course on Ordinary Differential Equations, we first
provide
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
3 / 33Ba
Course description
In this introductory course on Ordinary Differential Equations, we first
provide
basic terminologies on the theory of differential equations;
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
3 / 33Ba
Course description
In this introductory course on Ordinary Differential Equations, we first
provide
basic terminologies on the theory of differential equations;
proceed to methods of solving various types of ordinary differential
equations;
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
3 / 33Ba
Course description
In this introductory course on Ordinary Differential Equations, we first
provide
basic terminologies on the theory of differential equations;
proceed to methods of solving various types of ordinary differential
equations;
first order differential equations;
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
3 / 33Ba
Course description
In this introductory course on Ordinary Differential Equations, we first
provide
basic terminologies on the theory of differential equations;
proceed to methods of solving various types of ordinary differential
equations;
first order differential equations;
second and higher order linear differential equations;
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
3 / 33Ba
Course description
In this introductory course on Ordinary Differential Equations, we first
provide
basic terminologies on the theory of differential equations;
proceed to methods of solving various types of ordinary differential
equations;
first order differential equations;
second and higher order linear differential equations;
systems of differential equations;
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
3 / 33Ba
Course description
In this introductory course on Ordinary Differential Equations, we first
provide
basic terminologies on the theory of differential equations;
proceed to methods of solving various types of ordinary differential
equations;
first order differential equations;
second and higher order linear differential equations;
systems of differential equations;
discuss some related concrete mathematical modeling problems, which
can be handled by the methods introduced in this course;
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
3 / 33Ba
Course description
In this introductory course on Ordinary Differential Equations, we first
provide
basic terminologies on the theory of differential equations;
proceed to methods of solving various types of ordinary differential
equations;
first order differential equations;
second and higher order linear differential equations;
systems of differential equations;
discuss some related concrete mathematical modeling problems, which
can be handled by the methods introduced in this course;
Laplace transform if time permits.
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
3 / 33Ba
Textbooks
The main books are
Elementary Differential Equations and Boundary Value Problems,
Eleventh edition by William E. Boyce, Richard C. Diprima and Douglas
B. Meade.
Ordinary Differential Equation, by Morris Tenenbaum and Harry Pollard.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Other resources
Ordinary Differential Equations, Third Edition by Shepley L. Ross;
Ordinary Differential Equations, by Vladimir I. Arnold (for more
advanced students).
If necessary, you may use any other introductory level textbook on ordinary
differential equations.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Introduction
The study of differential equations has attracted the attention of many of the
world’s greatest mathematicians during the past three centuries. On the other
hand, it is important to recognize that differential equations remains a
dynamic field of inquiry today, with many interesting open questions.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Many of the principles, or laws, or processes underlying the behavior of the
natural world are statements or relations involving rates at which things
happen. When expressed in mathematical terms, the relations are equations
and the rates are derivatives. Equations containing derivatives are differential
equations. Therefore, to understand and to investigate problems involving the
motion of fluids, the flow of current in electric circuits, the dissipation of heat
in solid objects, the propagation and detection of seismic waves, or the
increase or decrease of populations, among many others, it is necessary to
know something about differential equations. A differential equation that
describes some physical process is often called a mathematical model of the
process, and many such models are discussed throughout this course.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Basic Concepts
Mathematical modeling of physical, economical, biological etc phenomena
often produces an equation which involves an ordinary or partial derivatives of
some unknown function. Such an equation is called a differential equation.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Basic Concepts
Mathematical modeling of physical, economical, biological etc phenomena
often produces an equation which involves an ordinary or partial derivatives of
some unknown function. Such an equation is called a differential equation.
A differential equation is an ordinary differential equation (ODE) if the
unknown function depends on only one independent variable.
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
8 / 33Ba
Basic Concepts
Mathematical modeling of physical, economical, biological etc phenomena
often produces an equation which involves an ordinary or partial derivatives of
some unknown function. Such an equation is called a differential equation.
A differential equation is an ordinary differential equation (ODE) if the
unknown function depends on only one independent variable.
If the unknown function depends on two or more independent variables, the
differential equation is a partial differential equation (PDE). The primary
focus of this course will be ordinary differential equations.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
8 / 33Ba
Basic Concepts
Mathematical modeling of physical, economical, biological etc phenomena
often produces an equation which involves an ordinary or partial derivatives of
some unknown function. Such an equation is called a differential equation.
A differential equation is an ordinary differential equation (ODE) if the
unknown function depends on only one independent variable.
If the unknown function depends on two or more independent variables, the
differential equation is a partial differential equation (PDE). The primary
focus of this course will be ordinary differential equations.
The order of a differential equation is the order of the highest derivative
appearing in the equation.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
Example 1.
Newton’s second low of motion applied to a free falling body leads to an ODE
dv
d2 h
= −mg.
=m
dt2
dt
where m is the mass of the object, g the gravitational acceleration, h the
height of the object, and v = dh
dt the velocity.
m
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
Example 1.
Newton’s second low of motion applied to a free falling body leads to an ODE
dv
d2 h
= −mg.
=m
dt2
dt
where m is the mass of the object, g the gravitational acceleration, h the
height of the object, and v = dh
dt the velocity.
m
Example
Example 2.
Modeling of vibrating strings, under some ideal conditions, leads to a PDE,
called the wave equation
2
∂2u
2∂ u
−
c
= 0.
∂t2
∂x2
where t is the time, x the location along the string, c the wave speed, and
u = u(x, t) the displacement of the string.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Simplest PDE examples
A few examples of second order linear PDEs in 2 variables are:
α2 uxx = ut
(one-dimensional heat conduction equation)
a2 uxx = utt
(one-dimensional wave equation)
uxx + uyy = 0 (two-dimensional Laplace/potential equation)
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
Example 3.
Equations (1) through (4) are examples, of ODEs, since the unknown function y
depends solely on the variable x. Equation (5) is a PDE, since y depends on both
the independent variables t and x.
dy
= 5x + 3
dx
2
d2 y
dy
ey 2 + 2
=1
dx
dx
d3 y
d2 y
4 3 + (sinx) 2 + 5xy = 0
dx
dx
2 3
7
2
dy
dy
d y
3
+
3y
+
y
= 5x
dx2
dx
dx
∂2y
∂2y
−4 2 =0
2
∂t
∂x
(1)
(2)
(3)
(4)
(5)
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
Example 3.
Equations (1) through (4) are examples, of ODEs, since the unknown function y
depends solely on the variable x. Equation (5) is a PDE, since y depends on both
the independent variables t and x.
dy
= 5x + 3
dx
2
d2 y
dy
ey 2 + 2
=1
dx
dx
d3 y
d2 y
4 3 + (sinx) 2 + 5xy = 0
dx
dx
2 3
7
2
dy
dy
d y
3
+
3y
+
y
= 5x
dx2
dx
dx
∂2y
∂2y
−4 2 =0
2
∂t
∂x
(1)
(2)
(3)
(4)
(5)
Example
Example 4. Equation (1) is a first-order DE; (2),(4), and (5) are second-order DE.
Equation (3) is a third-order DE.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Notation
The expressions y 0 , y 00 , y 000 , y 4 , ..., y n are often used to represent, respectively,
the first, second, third, fourth, ..., nth derivatives of y with respect to the
d2 y
independent variable under consideration. Thus, y represents dx
2 if the
2
independent variable is x, but represents ddpy2 if the independent variable is p.
Observe that parentheses are used in y (n) to distinguish it from the nth power,
y n . If the independent...variable is time, usually denoted by t, primes are often
replaced by dots ẏ, ÿ, y , ....
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Linear ODEs
Definition
A linear ordinary differential equation of order n, in the dependent variable y and the
independent variable x, is an equation that is in, or can be expressed in, the form
a0 (x)
dn−1 y
dy
dn y
+ a1 (x) n−1 + · · · + an−1 (x)
+ an (x)y = b(x)
n
dx
dx
dx
where a0 is not identically zero.
Observe
that the dependent variable y and its derivatives occur to the first degree only;
that no products of y and/or any of its derivatives are present;
that no transcendental functions of y and/or its derivatives occur.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
Example 5.
The following ordinary differential equations are both linear. In each case y is
the dependent variable. Observe that y and its various derivatives occur to the
first degree only and that no products of y and/or any of its derivatives are
present.
dy
d2 y
+5
+ 6y = 0
2
dx
dx
(6)
3
d4 y
dy
2d y
+
x
+ x3
= xex
4
3
dx
dx
dx
(7)
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Linear ordinary differential equations are further classified according to the
nature of the coefficients of the dependent variables and their derivatives. For
example, Equation (6) is said to be linear with constant coefficients, while
Equation (7) is linear with variable coefficients.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Nonlinear ODEs
Definition
A nonlinear ordinary differential equation is an ordinary differential equation that is
not linear.
Example
Example 6.
The following ordinary differential equations are all nonlinear:
d2 y
dy
+ 6y 2 = 0
+5
dx2
dx
3
d2 y
dy
+
5
+ 6y = 0
dx2
dx
d2 y
dy
+ 5y
+ 6y = 0
dx2
dx
(8)
(9)
(10)
Equation (8) is nonlinear because the dependent variable y appears to the second
degree in the term 6y 2 . Equation (9) owes its nonlinearity to the presence of the
term 5(dy/dx)3 , which involves the third power of the first derivative. Finally,
equation (10) is nonlinear because of the term 5y(dy/dx), which involves the product
of theDukenbayeva
dependenta.dukenbayeva@sdu.edu.kz
variable and
its first
derivative.
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Ordinary
Differential
Equations Lecture 1: Introduction and direction fields.
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Application of DE
Differential equations occur in connection with numerous problems that are
encountered in the various branches of science and engineering. For example
The problem of determining the motion of a projectile, rocket, satellite, or
planet.
The problem of determining the charge or current in an electric circuit.
The problem of the conduction of heat in a rod or in a slab.
The problem of determining the vibrations of a wire or a membrane.
The study of the rate of decomposition of a radioactive substance or the
rate of growth of a population.
The study of the reactions of chemicals.
The problem of the determination of curves that have certain geometrical
properties.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Solutions
A solution of a differential equation in the unknown function y and the independent
variable x on the interval I, is a function y(x) that satisfies the differential equation
identically for all x in I.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Solutions
A solution of a differential equation in the unknown function y and the independent
variable x on the interval I, is a function y(x) that satisfies the differential equation
identically for all x in I.
Example
Example 7.
Is y(x) = c1 sin 2x + c2 cos 2x, where c1 and c2 are arbitrary constants, a solution of
y 00 + 4y = 0? Differentiating y, we find
y 0 = 2c1 cos 2x − 2c2 sin 2x and y 00 = −4c1 sin 2x − 4c2 cos 2x
Hence, y 00 + 4y = (−4c1 sin 2x − 4c2 cos 2x) + 4 (c1 sin 2x + c2 cos 2x)
= (−4c1 + 4c1 ) sin 2x + (−4c2 + 4c2 ) cos 2x
=0
Thus, y = c1 sin 2x + c2 cos 2x satisfies the differential equation for all values of x and
is a solution on the interval (−∞, ∞).
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
Example 8.
Determine whether y = x2 − 1 is a solution of (y 0 )4 + y 2 = −1.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
Example 8.
Determine whether y = x2 − 1 is a solution of (y 0 )4 + y 2 = −1.
Note that the left side of the differential equation must be nonnegative for
every real function y(x) and any x, since it is the sum of terms raised to the
second and fourth powers, while the right side of the equation is negative.
Since no function y(x) will satisfy this equation, the given differential equation
has no solution.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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We see that some differential equations have infinitely many solutions
(Example 7), whereas other differential equations have no solutions (Example
8). It is also possible that a differential equation has exactly one solution.
Consider (y 0 )4 + y 2 = 0, which for reasons identical to those given in Example
8 has only one solution y = 0.
It is customary to call a solution which contains n constants c1 , c2 , ..., cn an
n-parameter family of solutions, and to refer to the constants c1 to cn as
parameters.
For the classes of differential equations we shall consider, we can now assert: a
differential equation of the nth order has an n-parameter family of solutions.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Definition
A solution of a differential equation will be called a particular solution if it
satisfies the equation and does not contain arbitrary constants.
Definition
An n-parameter family of solutions of differential equation will be called a
general solution if it contains every particular solution.
So a particular solution of a differential equation is any one solution. The
general solution of a differential equation is the set of all solutions.
Example
The general solution to the differential equation in Example (9) can be shown
to be y = c1 sin 2x + c2 cos 2x. That is, every particular solution of the
differential equation has this general form. A few particular solutions are:(a)
y = 5 sin 2x − 3 cos 2x (choose c1 = 5 and c2 = −3), (b) y = sin 2x (choose
c1 = 1 and c2 = 0 ), and (c) y = 0 (choose c1 = c2 = 0).
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Initial-value and boundary-value problems
We shall begin this section by considering the rather simple problem of the following
example.
Example
Find a solution f of the differential equation
dy
= 2x
dx
(11)
such that at x = 1 this solution f has the value 4.
Explanation. We seek a real function f which fulfills the two following
requirements:
The function f must satisfy the differential equation (11). That is the function
f must be such that f 0 (x) = 2x for all real x in a real interval I.
The function f must have the value 4 at x = 1. That is, that function f must
be such that f (1) = 4.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Solution. We observe that the differential equation has one-parameter family of
solutions
y = x2 + c,
(12)
where c is an arbitrary constant. Let us now determine the constant c so that (12)
satisfies requirement 2, that is y(1) = 4. By simpel calculation, we find that
f (x) = x2 + 3.
If all of the associated supplementary conditions relate to one x value, the problem is
called an initial-value problem (or one-point boundary-value problem).
If the condition relate to two different x values, the problem is called a two-point
boundary-value problem (or simply a boundary-value problem)
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
d2 y
+y =0
dx2
y(1) = 3,
y 0 (1) = −4.
Both of this conditions relate to one x value, namely, x = 1. Thus, this is an
initial-value problem.
Example
d2 y
+y =0
dx2
y(0) = 1,
π
y( ) = 5.
2
The conditions relate to the two different x values, 0 and
boundary-value problem.
π
.
2
Thus, this is a
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Now we turn to a more detailed consideration of the initial-value problem for
a first-order differential equation.
Definition
Consider the first-order differential equation
dy
= f (x, y)
dx
(13)
where f is a continuous function of x and y in some domain D of the xy
plane; and let (x0 , y0 ) be a point of D. The initial-value problem associated
with (13) is to find a solution φ of the differential equation (13), defined on
some real interval containing x0 , and satisfying the initial condition
φ (x0 ) = y0
In the customary abbreviated notation, this initial-value problem may be
written
dy
= f (x, y)
dx
y (x0 ) = y0
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Geometric interpretation of the initial-value problem
To solve this problem, we must find a function φ that not only satisfies differential
equation (13) but that also satisfies the initial condition that it has the value y0
when x has value x0 . The geometric interpretation of the initial condition, and hence
of the entire initial-value problem, is easily understood. The graph of the desired
solution φ must pass through the point with coordinates (x0 , y0 ). That is,
interpreted geometrically, the initial-value problem is to find an integral curve of the
differential equation (13) that passes through the point (x0 , y0 ).
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
Solve the initial-value problem
dy
x
=−
dx
y
(14)
y(3) = 4
(15)
given that the DE (14) has a one-parameter family of solutions
x2 + y 2 = c 2
(16)
The pair of values (3, 4) must satisfy the relation (16). Substituting x = 3 and y = 4 into
(16), we find
9 + 16 = c2 or c2 = 25
Now substituting this value of c2 into (16), we have
x2 + y 2 = 25
Solving this for y, we obtain
y=±
p
25 − x2
Obviously the positive sign must be chosen to give y the value +4 at x = 3. Thus the
function f defined by
p
f (x) = 25 − x2 , −5 < x < 5
is the solution of the problem.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Existence of Solutions
Do all initial-value and boundary-value problems have solutions? We can
easily check that the following boundary-value problem
d2 y
+y =0
dx2
y(0) = 1
y(π) = 5
has no solution! Thus arises the question of existence of solutions: given an
initial-value or boundary-value problem, does it actually have a solution?
Every initial-value problem that satisfies the definition given above has at least
one solution.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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But now another question is suggested, the question of uniqueness. Does such
a problem ever have more than one solution? Let us consider the initial-value
problem
dy
= y 1/3
dx
y(0) = 0
One may verify that the functions f1 and f2 defined, respectively, by
f1 (x) = 0
and
f2 (x) =
2
x
3
for all real x
3/2
,
x ≥ 0;
f2 (x) = 0,
x≤0
are both solutions of this initial-value problem! In fact, this problem has
infinitely many solutions! The answer to the uniqueness question is clear: the
initial-value problem, as stated, need not have a unique solution. In order to
ensure uniqueness, some additional requirement must certainly be imposed.
We shall see that in the following theorem.
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Basic existence and uniqueness theorem
Hypothesis. Consider the differential equation
dy
= f (x, y)
dx
(17)
where
1. The function f is a continuous function of x and y in some domain D of the
xy plane, and
2. The partial derivative ∂f /∂y is also a continuous function of x and y in D;
and let (x0 , y0 ) be a point in D.
Conclusion. There exists a unique solution φ of the differential equation
(17). defined on some interval |x − x0 | ≤ h, where h is sufficiently small, that
satisfies the condition
φ (x0 ) = y0
(18)
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Explanation of the theorem
1. It is an existence and uniqueness theorem. This means that it is a theorem which
tells us that under certain conditions (stated in the hypothesis) something exists and
is unique. It gives no hint whatsoever concerning how to find this solution but
merely tells us that the problem has a solution.
2. The hypothesis tells us what conditions are required of the quantities involved. It
deals with two objects: the differential equation (17) and the point (ẋ0 , y0 ). As far
as the differential equation (17) is concerned, the hypothesis requires that both the
function f and the function ∂f /∂y must be continuous in some domain D of the xy
plane. As far as the point (x0 , y0 ) is concerned, it must be a point in this same
domain D, where f and ∂f /∂y are so well behaved (that is, continuous).
3. The conclusion tells us of what we can be assured when the stated hypothesis is
satisfied. It tells us that we are assured that there exists one and only one solution φ
of the differential equation, which is defined on some interval |x − x0 | ≤ h centered
about x0 and which assumes the value y0 when x takes on the value x0 . That is, it
tells us that, under the given hypothesis on f (x, y), the initial-value problem
dy
= f (x, y)
dx
y (x0 ) = y0
has a unique solution that is valid in some interval about the initial point x0 .
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Ordinary Differential Equations Lecture 1: Introduction and direction fields.
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Example
Consider the initial-value problem
dy
= x2 + y 2
dx
y(1) = 3
Let us apply Theorem 1.1. We first check the hypothesis. Here
(x,y)
= 2y. Both of the functions f and ∂f /∂y are
f (x, y) = x2 + y 2 and ∂f ∂y
continuous in every domain D of the xy plane. The initial condition y(1) = 3
means that x0 = 1 and y0 = 3, and the point (1, 3) certainly lies in some such
domain D. Thus all hypotheses are satisfied and the conclusion holds. That is,
there is a unique solution φ of the differential equation dy/dx = x2 + y 2 ,
defined on some interval |x − 1| ≤ h about x0 = 1, which satisfies that initial
condition, that is, which is such that φ(1) = 3
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
32 / 33Ba
End of Lecture 1
Aishabibi Dukenbayeva a.dukenbayeva@sdu.edu.kz
Ordinary Differential Equations Lecture 1: Introduction and direction fields.
33 / 33Ba