Download Lecture 1. Introduction

Ordinary and Partial Differential Equations
Lecture 1. Introduction
Bibliography: Nagle, Snaff, Sieder - ”Fundamentals of DEs and BVPs” (6th ed.) - sections 1.1, 1.2, 1.3
BACKGROUND
In the sciences and engineering, mathematical models are developed to aid in the understanding of
physical phenomena, often yielding an equation that contains some derivatives of an unknown function.
Such an equation is called a differential equation.
Example 1: Free fall of a body.
When an object is released from a certain height above the ground and falls under the force of gravity,
Newtons second law (which states that an objects mass times its acceleration equals the total force acting
on it) can be applied to the falling object. This leads to the differential equation:
d2 h
= −mg
(1)
dt2
where m is the mass of the body, g is the gravitational acceleration, h = h(t) is the height above the
d2 h
ground and the second order derivative 2 is the acceleration of the body, at a certain moment of time t.
dt
−gt2
♣ Solving the equation (1) leads us to the solution: h(t) =
+ c1 t + c2 , where the constants c1 and
2
c2 may be determined from the initial height and initial velocity of an object.
m·
Example 2: Radiocative decay.
We assume that the rate of decay is proportional to the amount of radioactive substance present. This
leads to the differential equation:
dA
= −kA
(2)
dt
where A = A(t) is the unknown amount of radioactive substance at time t and k is the proportionality
dA
constant. The first-order derivative
represents the rate of decay of the radioactive substance at time t.
dt
♣ The solution of equation (2) is A(t) = Ce−kt , where the constant C is the initial amount of radioactive
substance.
→ Whenever a mathematical model involves the rate of change of one variable with respect to another, a
differential equation is apt to appear. Unfortunately, in contrast to the previous examples, the differential
equation may be very complicated and difficult to analyze.
z See other examples of differential equations arising from different subject areas on page 3 of [Nagle].
TERMINOLOGY
• If an equation involves the derivative of one variable with respect to another, then the former is
called a dependent variable and the latter an independent variable.
• A differential equation involving only ordinary derivatives with respect to a single independent
variable is called an ordinary differential equation (ODE).
• A differential equation involving partial derivatives with respect to more than one independent
variable is a partial differential equation (PDE).
• The order of a DE is the order of the highest-order derivatives present in the equation.
• A linear ordinary differential equation is an ODE of the form:
an (x)
dn−1 y
dy
dn y
+
a
(x)
+ ... + a1 (x)
+ a0 y(x) = G(x)
n−1
dxn
dxn−1
dx
• If an ODE is not linear, then we call it nonlinear.
c 2011/2012 Eva Kaslik - West University of Timişoara, Romania
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SOLUTIONS AND INITIAL VALUE PROBLEMS
The general form for an nth-order ODE with x independent and y dependent variable, is written as
(
)
dy
dn y
F x, y, , . . . , n = 0
, ∀ x ∈ (a, b)
(3)
dx
dx
where F is a real valued function of (n + 2) real variables, and the endpoints of the interval (a, b) may
be infinite.
dn y
In many cases, the highest-order term
can be isolated, and the ODE (3) can be rewritten as
dxn
)
(
dn y
dn−1 y
dy
,
.
.
.
,
=
f
x,
y,
(4)
dxn
dx
dxn−1
which is often preferable to (3) for theoretical and computational purposes.
♣ A function ϕ(x) that when substituted for y in equation (3) [or (4)] satisfies the equation for all x in
the interval I = (a, b) is called an explicit solution to the equation on I.
♣ A relation G(x, y) = 0 is said to be an implicit solution to equation (3) on the interval I = (a, b) if
it defines one or more explicit solutions on I.
z See examples on pages 6-8 of [Nagle].
♣ By an initial value problem (IVP) for an nth-order differential equation (3) [or (4)] we mean:
Find a solution to the differential equation on an interval I that satisfies at x0 the n initial conditions:
y(x0 ) = y0
dy
(x0 ) = y1
dx
..
.
dn−1 y
(x0 ) = yn−1
dxn−1
where x0 ∈ I and y0 , y1 , ..., yn−1 are given constants.
→ In the case of a first-order equation, the initial conditions reduce to the single requirement: y(x0 ) = y0 .
z See examples on pages 10-11 of [Nagle].
Existence and Uniqueness Theorem for first-order initial value problems:
Consider the initial value problem
dy
= f (x, y)
dx
,
y(x0 ) = y0 .
(5)
df
are continuous functions on some rectangle R = (a, b) × (c, d) that contains the point (x0 , y0 ),
dy
then the IVP (5) has a unique solution ϕ(x) in some interval (x0 − δ, x0 + δ), where δ > 0.
If f and
DIRECTION FIELDS
For many DEs encountered in real-world applications, it is impossible to find an explicit solution.
However, for practical reasons, we may need to know something about the nature of the solution, such as
the intervals where the solution is increasing, or the points where the solution attains a maximum value.
One technique that is useful in visualizing the solutions to a first-order differential equation is to sketch
the direction field for the equation. A first-order ODE
dy
= f (x, y)
dx
specifies a slope at each point in the xy-plane where f is defined. In other words, it gives the direction
that a graph of a solution to the equation must have at each point.
♣ A plot of short line segments drawn at various points in the xy-plane showing the slope of the solution
curve there is called a direction field for the differential equation.
z Method of isoclines - page 19 in [Nagle].
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