TERM PAPER

ENGINEERING MATHEMATICS-I
MTH101

TOPIC: Geometrical Applications of Ordinary Differential

Equations

Date of Allotment: 14/09/2010

Date of Submission: 18/11/2010

Submitted to: Submitted by:

Ms. Reetika Salaria Mr. Nand Kishore Singh

Deptt. of Mathematics Roll No: RD1001A14

Regd.No: 11007673

Section: D1001
 ACKNOWLEDGEMENT

I NAND KISHORE SINGH, student of B.Tech-M.Tech (CSE)

(Section-D1001) express my deep gratitude Ms. Reetika
Salaria. I am thankful to her for support that led me to the
completion of this term paper.
I am very thankful to my parents who encouraged me
and provided me all the necessary resources that had made
possible for me to be able to accomplish this task.
I also thank all my friends who assisted me in completing this
work.
CONTENTS

1. Introduction to Differential Equations

2. Nomenclature
3. Applications of Differential Equations
Introduction mostly concerned with their solutions, the
set of functions that satisfy the equation.
A differential equation is a mathematical Only the simplest differential equations
equation for an unknown function of one admit solutions given by explicit formulas;
or several variables that relates the values however, some properties of solutions of a
of the function itself and its derivatives of given differential equation may be
various orders. Differential equations play determined without finding their exact
a prominent role in engineering physics, form. If a self-contained formula for the
economics and other disciplines. solution is not available, the solution may
Differential equations arise in many areas be numerically approximated using
of science and technology: whenever a computers. The theory of dynamical
deterministic relationship involving some systems puts emphasis on qualitative
continuously varying quantities (modeled analysis of systems described by
by functions) and their rates of change in differential equations, while many
space and/or time (expressed as numerical methods have been developed to
derivatives) is known or postulated. This is determine solutions with a given degree of
illustrated in classical mechanics, where accuracy.
the motion of a body is described by its
Nomenclature
position and velocity as the time varies.
Newton’s laws allow one to relate the The theory of differential equations is
position, velocity, acceleration and various quite developed and the methods used to
forces acting on the body and state this study them vary significantly with the type
relation as a differential equation for the of the equation.
unknown position of the body as a  An ordinary differential equation
function of time. In some cases, this (ODE) is a differential equation in
differential equation (called an equation of which the unknown function (also
motion) may be solved explicitly. known as the dependent variable)
An example of modeling a real world is a function of single independent
problem using differential equations is variable. In the simplest form, the
determination of the velocity of ball falling unknown function is a real or
through the air, considering only gravity complex valued function, but more
and air resistance. The ball’s acceleration generally, it may be vector-valued
towards the is the acceleration due to or matrix-valued: this corresponds
gravity minus the deceleration due to air to considering a system of ordinary
resistance. Gravity is constant but air differential equations for a single
resistance may be modeled as proportional function. Ordinary differential
to the ball’s velocity. This means the ball’s equations are further classified
acceleration, which is the derivative of its according to the order of the
velocity, depends on the velocity. Finding highest derivative with respect to
the velocity as a function of time requires the dependent variable appearing in
solving a differential equation. the equation. The most important
cases for applications are first order
Differential equations are mathematically
and second order differential
studied from several different perspectives,
 equations. In the classical literature differential equation are allowed to be
also distinction is made between (known) functions of the independent
differential equations explicitly variable or variables; if these coefficients
solved with respect to the highest are constants then one speaks of a constant
derivative and differential coefficient linear differential equation.
equations in an implicit form. There are very few methods of explicitly
 A partial differential equation solving non-linear differential equations;
(PDE) is a differential equation in those that are known typically depend on
which the unknown function is a the equation having particular symmetries.
function of multiple independent Non-linear differential equations can
variables and the equation involves exhibit very complicated behavior over
its partial derivatives. The order is extended time intervals, characteristic of
defined similarly to the case of chaos. Even the fundamental questions of
ordinary differential equations, but existence, uniqueness, and extendability of
further classification into elliptic, solutions for non-linear differential
hyperbolic, and parabolic equations, and well-posedness of initial
equations, especially for second and boundary value problems for non-
order linear equations, is of utmost linear PDEs are difficult problems and
importance. Some partial their resolution in special cases is
differential equations do not fall considered to be a significant advance in
into any of these categories over the mathematical theory (cf Navier-Stokes
the whole domain of the existence and smoothness).
independent variables and they are
Linear differential equations frequently
said to be of mixed type.
appear as approximations to non-linear
equations. These approximations are only
Both ordinary and partially differential valid under restricted conditions. For
equations are broadly classified as linear example, the harmonic oscillator equation
and non-linear. A differential equation is is an approximation to the non-linear
linear if the unknown function and its pendulum equation that is valid for small
derivatives appear to the power 1 amplitude oscillations.
(products are not allowed) and non-linear
Examples:
otherwise. The characteristic property of
linear equations is that their solutions form In the first group of examples, let u be an
an affine subspace of an appropriate unknown function of x, and c, ω are
function space, which results in much known constants.
more developed theory of linear  In homogeneous first order
differential equations. Homogenous linear linear constant coefficient
differential equations are a further subclass ordinary differential equation:
for which the space of solutions is a linear d2u = cu + x2
subspace i.e. the sum of any set of dx
solutions or multiples of solutions is also a  Homogeneous second order
solution. The coefficients of the unknown linear ordinary differential
function and its derivatives in a linear equation:
 d2u - x du + u= 0  Third order non-linear partial
dx2 dx differential equation, the
 Homogeneous second order Korteweg-de Vries equation:
constant coefficient linear
ordinary differential equation
describing the harmonic
oscillator:
Applications of Differential
d2u + ω2u =0
Equations
dx2
We present examples where differential
equations are widely applied to model
 First order non-linear ordinary natural phenomena, engineering systems
differential equation: and many other situations.
du = u2 + 1
dx
Radioactive Decay
 Second order non-linear Many radioactive materials disintegrate at
ordinary differential equation a rate proportional to the amount present.
describing the motion of a For example, if X is the radioactive
pendulum of length L: material and Q(t) is the amount present at
time t, then the rate of change of Q(t) with
respect to time t is given by
In the next group of examples, the dQ = -rQ,
unknown function u depends on two dt
variables x and t or x and y.
 Homogeneous first order linear where r is a positive constant (r>0). Let us
partial differential equation: call Q(0)-Q0 the initial quantity of the
material X, then we have,
Q(t) = Q0e-rt.
Clearly in order to determine Q(t) we need
to find the constant r. This can be done
 Homogeneous second order using what is called the half life T of the
linear constant coefficient material X. The half life is the time span
partial differential equation of needed to disintegrate half of the material.
elliptic type, the Laplace So, we have Q(T)= ½ Q0. An easy
equation: calculation gives rT= ln(2). Therefore, if
we know T, we can get r and vice-versa.

Newton’s Law of Cooling

From experimenatal observations it
is known that (up to a “satisfactory”
approximation) the surface
temperature of an object changes at implicit equation with a parameter
a rate proportional to its relative something like
temperature. That is, the difference F(y,t) = C
between its temperature and the This is an equation describing a family of
temperature of the surronding curves. Whenever we fix the parameter C
environment. This is what is known we get one cuurve and vice-versa. For
example, consider the families of curves
as Newton’s law of cooling. Thus, if
ϴ(t) is the temperature of the object y-mt or y2 + t2 – C2
at time t, then we have where m and C are parameters. Clearly, we
may change the names of the variables and
dϴ = -k(ϴ - S) still have the same geometric curves. For
dt example, the above families define the
same geometric object as
where S is the temperature of the
y –mx or y2 + x2 – C2
surrounding environment. A
qualitative study of this phenomena Note that the first family descrbes all the
lines passing thrugh the origin (0,0) whi,e
will show that k>0. This is a first
the second family describes alll the circles
order linear differential equation. centred at the origin (including all the limit
The solution, under the initial case when the radius 0 reduces to the
condition ϴ(0) = ϴ0, is given by single point (0,0).
ϴ(t) =S + (ϴ0 – S)e-rt. In this we will only use the variables x and
y. Any family of curves will be written as
Hence,
F(x,y,C) = 0
-k(t1-t2)
ϴ(t1) – S = e
One may ask whether any family of curves
ϴ(t2) – S may be generated from a differeential
which implies equation? In geeneral, the answer is no.
Let us see how to proceed if the answer
k(t1 – t2) = - ln ( ϴ(t1) – S) were to be yes. First differentiate with
ϴ(t2) – S respect to x, and get a new equation
involving in general x, y, dy , and C. Using
This equation makes it possible to
dx
find k if the interval of time ( t 1 – t2)
the original equation, we may able to
is known and vice-versa. eliminate the parameter C from the new
equation.

Orthogonal Trajectories Definition of Orthogonal Curves:

Consider two families of curves f1 and f2.
We say that f1 and f2 are orthogonal
whenever any curve from f1 intersects any
We have seen before that the solutions of a
curve from f2, the two curves are
differential equation may be given by an orthogonal at the point of intersection.
For example, we have seen that the
families y= mx and x2+ y2 = C2 are Step 3. Write down the
orthogonal. differential equation associated to the
Given a family of curves f, is it possible orthogonal family
to find a family of curves which is dy = -1
orthogonal to f? dx f(x, y)
The answer to this question has many Step 4. Solve the new equation.
implications in many areas such as The solutions are exactly the family
physics, fluid dynamics, etc. In general,
of orthogonal curves.
this question is very difficult. But in some
cases, we may be able to carry on the Step 5. You may be asked to
calculations and find the orthogonal give a geometric view of the two
family. Let us show how: families. Also you may be asked to
Consider the family of curves f. We find a specific curve from the
assume that an associated differential orthogonal family.
equation may be found, say
Population Dynamics
dy = f(x,y)
dx Here are some natural questions
We know that for any curve from the related to population problems:
family passing by the point (x, y), the
slope of the tangent at this point is f(x, y).  What will the population of a
Hence the slope of the line perpendicular certain country be in ten years?
(or orthogonal) to this tangent is  How are we protecting the
-1 resources from extinction?
f(x, y) which happens to be the slope
of the tangent line to the orthogonal More can be said about the problem
but, in this little review we will not
curve passing by the point (x, y). In
discuss them in detail. In order to
other words the family of orthogonal illustrate the use of differential
curves are solutions to the equations with regard to this problem
differential equation we consider the easiest mathematical
dy = -1 model offered to govern the
dx f(x, y) population dynamics of a certain
species. It is commonly called the
exponential model, that is, the rate
From this we see what we have to
of change of the population is
do. Indeed consider a family of
proportional to the existing
curves f. In order to find the
population. In other words, if P(t)
orthogonal family, we use the
measures the population, we have
following practical steps
dP = kP
Step 1. Find the associated
dt
differential equation.
Step 2. Rewrite the differential where the rate k is constant. It is
equation in explicit form. fairly easy to see that if k>0, we have
dy = f(x, y) growth, and if k<0, we have decay.
dx
 This is a linear equation which solves and integration
into

P(t) = P0ekt, P
∫ dP/P (1− M ) = ∫ kdt.
Where P0 is the initial population,
i.e. P(0) = P0. Therefore, we The partial fraction techniques gives
conclude the following: 1
 if k>0, then the population grows
and continues to expand to infinity, ∫ dP/P 1− MP = ∫ ( P1 + MP )dP
( )
that is, 1−
M
lim P(t) = +∞; ,
t→+∞ which gives
P
 if k<0, then the population will ln |P|−¿ ln ¿ 1− ∨¿ ¿ ¿ = kt + c.
M
shrink and tend to 0. In other words
we are facing extinction. Easy algebraic manipulations give
Clearly, the first case, k>0, is not adequate P
= Cekt,
and the model can be dropped. The main 1−P/ M
argument for this has to do with where C is a constant. Solving for P, we
environmental limitations. The get
complication is that population growth is
MCekt
eventually limited by some factor, usually P= .
one from among many essential resources. M +Cekt
When a population is far from its limits of If we consider the initial condition P(0) =
growth it can grow exponentially. P0 (assuming that P0 is not equal to both 0
However, when nearing its limits the or M), we get
population size can fluctuate, even C = P0M
chaotically. Another model was proposed
to remedy this flaw in the exponential M- P0,
model. It is called the logical model (also which, once substituted into the expression
called Verhulst-Pearl model). The for P(t) and simplified, we find
differential equation for this model is
P (t) = MP0
dP P
= kP ( 1 - ). P0 + (M-P0)e-kt
dt M
It is easy to see that
where M is a limiting size for the
population (also called the carrying limP(t)=M
capacity). Clearly, when P is small t→+∞
compared to M, the equation reduces to
However, this is still not satisfactory
the exponential one. In order to solve this
because this model does not tell us when a
equation we recognize a non-linear
population is facing extinction since it
equation which is separable. The constant
never implies that. Even starting with a
solutions are P=0 and P=M. The non-
small population it will always tend to the
constant solutions may obtained by
carrying capacity M.
separating the variables
dP Some other Applications to
P = kdt Engineering and Sciences:
P (1− )
M
Historically, it has been the needs of Historically, the source of Fourier
the physical sciences which have series.
driven the development of many  Quantum theory studies the
solutions of the Schrodinger
parts of mathematics , particularly (differential) equation. Also
analysis. The applications are includes a good deal of Lie group
sometimes difficult to classify theory and quantum group theory,
mathematically, since tools from theory of distributions and topics
several areas of mathematics may be from Functional analysis, Yang-
applied. We focus on these Mills problems, Feynman
diagrams, and so on
applications not by discussing the  Statistical mechanics, structure
nature of their discipline but rather of matter is the study of large
their interaction with mathematics. scale systems of particles,
 Mechanics of particles and including stochastic systems and
systems studies dynamics of sets of moving or evolving systems.
particles or solid bodies, including Specific types of matter studied
rotating and vibrating bodies. Uses include fluids, crystals, metals, and
variational principles (energy other solids.
minimization) as well as  .Relativity and gravitational theory
differential equations. is differential geometry, analysis,
 Mechanics of deformable solids and group theory applied to physics
considers questions of elasticity on a grand scale or in extreme
and plasticity, wave propagation, situations (e.g black holes, and
engineering, and topics in specifics cosmology).
solids such as soils and crystals.  Astronomy and astrophysics : as
 Fluid mechanics studies air, water, celestial mechanics is,
and other fluids in motion: mathematically, part of Mechanics
compression, turbulence, diffusion, of particles, the principle
wave propagation, and so on. applications in this area appear to
Mathematically this includes study be concerning the structure,
of solutions of differential evolution, and interaction of stars
equations, including large-scale and galaxies.
numerical methods (e.g the finite-  Geophysics applications typically
element method). involve material in mechanics and
 Optics, electromagnetic theory is fluid mechanics, as above, but for
the study of the propagation and large scale problems (this deals
evolution of electromagnetic with a very big solid and a large
waves, including topics of pool of fluid).
interference and diffraction.  Systems theory; control study the
Besides the usual branches of evolution over time of complex
analysis, this area includes systems such as those in
geometric topics such as the paths engineering. In particular, one may
of light rays. try to identify the system- to
 Classical thermodynamics, heat determine the equations or
transfer is the study of the flow of parameters which govern its
heat through matter, including development – or to control the
phase change and combustion. system – to select the parameters
(e.g. via feedback loops) to achieve
a desired state. Of particular
 interest are issues in stability
( steady state configurations) and
the effects of random changes and
noise( stochastic systems). While
popularly the domain of
“cybernetics” or “robotics”,
perhaps, this is in practice a field of
application of differential (or
difference) equations, functional
analysis, numerical analysis, and
global analysis (or differential
geometry).
Observe that the branches of
mathematics most closely allied with
the fields of mathematical physics are
the parts of analysis, particularly those
related to differential equations. The
other sciences draw on these as well as
probability and statistics and,
increasingly, numerical methods.