Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
Q. 4(a) Find power series solution of (1 + x2 )y + xy y = 0 about x = 0.
Solution: Comparing the given dierential equation with y + p(x)y + q(x)y = 0, we get
p(x) =
x
1 + x2
and q(x) =
1
.
1 + x2
We see that p(x) and q(x) are analytic at x = 0. Therefore,
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
HOMOGENEOUS LINEAR
SYSTEM WITH CONSTANT
COEFFICENTS:
TWO EQUATIONS IN TWO
UNKNOWN FUNCTIONS
20052006, MATH C241 Prepared by NadeemurRehman
11/12/2010
1
We Shall concerned with the homogeneous
linear system
dx
a1 x b1 y
dt
dy
a2 x b2 y
dt
(1)
where
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Qualitative Properties of solutions of a second order
homogeneous Linear Differential equations.
Throughout this chapter we shall be
looking at the second order homogeneous
linear differential equation
y P( x) y Q( x) y 0 .(1)
We shall like to say somethi
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
FROBENIUS SERIES
SOLUTION OF A SECOND
ORDER HOMOGENEOUS
LINEAR DIFFERENTIAL
EQUATION (C0NTINUED)
Assume that x = 0 is a regular singular point
of the second order homogeneous l.d.e.
y P( x) y Q( x) y 0
Hence
p( x) xP( x)
q( x) x Q( x)
2
are both analytic
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Laplace Transform
12.11.2010
20052006, NadeemurRehman
1
In this chapter we will introduce a technique
as the Laplace transform, which is a very
useful tool in the study of linear differential
equations. Although be no means limited to
this class of pro
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Fourier Series
20052006, NadeemurRehman
12.11.2010
We will see that many important problems involving
partial differential equations can be solved,
provided a given function can be expressed as an
infinite sum of sines and cosines.
In this section and
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
PARTIAL DIFFERENTIAL EQUATIONS:
THE VIBRATING STRING:
Suppose that a flexible string is pulled taut on
the xaxis and fastened at two points, which we
denote by x = 0 and x = L. The string is then
drawn aside into a certain curve y = f(x) in the
xy plane
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Review of Power Series
Real Analytic functions
Ordinary points of a 2nd
Order Homogeneous L.D.E.
In this lecture we discuss the Power series
method of solving a second order
homogeneous linear differential equation
First we spend some time in reviewing th
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Inverse Laplace Transform:
If L[f(x)] = F(p), then f(x) is called an inverse
Laplace transform of F(p), and we write
f(x) = L1[F(p)].
1
1
1[ ] 1.
L[1] L
p
p
n ] n! L1[ n! ] x n
L[ x
n1
n1
p
p
1
xn
L1[
]
n!
p n1
12.11.2010
20052006, NadeemurRehman
1
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Even and Odd Functions
A function f, defined on an interval centred at
the origin is said to be even if
f ( x) f ( x)
For all x in the domain of f, and odd if
f ( x) f ( x)
The graph of even function is symmetric about the
yaxis. The graph of odd functio
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Bessels Differential
equation (continued)
12Nov10
MATH C241 Prepared by MSR
1
In this lecture we study properties of
Bessels functions, which are solutions of
Bessels equation.
12Nov10
MATH C241 Prepared by MSR
2
Zeros of Bessel Functions
Fact:
(i) If
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Bessels Differential
equation
12Nov10
MATH C241 prepared by MSR
1
In this lecture we discuss Bessels
Differential Equation. We also study
properties of Bessels functions, which are
solutions of Bessels equation. We first
review the definition and proper
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Some Special Functions of
Mathematical Physics
Legendre Polynomials
21 SEP 2005
1
There are many special functions that are
being used by Physicists in the solution of
physical problems, like, Legendre
Polynomials, Hermite Polynomials, Bessels
Functions,
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
LAPLACE TRANSFORMS
11/12/2010
MATH C241
Prepared by MSR
1
LAPLACE TRANSFORMS
Given a realvalued function f(x) defined for
all x 0, we define its Laplace transform as
L[ f ( x)] e
px
f ( x)dx F ( p)
0
where p is a real number.
11/12/2010
MATH C241
Prepar
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
GAUSSS
HYPERGEOMETRIC
EQUATION (Continued)
12Nov10
MATH C241 Prepared by MSR
1
Abstract
In this lecture we continue the study of
Hypergeometric equation. We also show
that certain second order homgeneous l.d.e.
can be transformed into a hypergeometric
e
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
GAUSSS
HYPERGEOMETRIC
EQUATION
12Nov10
MATH C241 Prepared by MSR
1
Abstract
In this lecture we discuss the famous 2nd
order homogeneous l.d.e. the so called
Gausss hypergeometric d.e. We also show
that many elementary functions are
solutions of suitable
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
20052006, NadeemurRehman
1
A system of simultaneous first
order ordinary differential
equations has the general
form
x1 F1 (t , x1, x2 ,
x2 F2 (t , x1, x2 ,
xn Fn (t , x1, x2 ,
xn )
xn )
xn )
20052006, NadeemurRehman
(1)
2
where each xk is a func
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
We assume a particular solution of
Ly k1e cos bx k2e sin bx
ax
ax
as y y p Ae cos bx A2e sin bx
1
ax
ax
We assume a particular solution of
ax
n
Ly e (b0 b1x . bn x )
as y y p e ( A0 A1x . An x )
We also use multiply by the least power of x
rule in case an
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
FIRST ORDER
DIFFERENTIAL EQUATIONS
In this lecture we discuss various methods
of solving first order differential equations.
These include:
Variables separable
Homogeneous equations
Exact equations
Equations that can be made exact by
multiplying by an
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
Q 3(a) Using operator method find particular integral of
[12]
Solution: The associated homogeneous equation is
Auxiliary equation is
Thus
Now
[2M]
[2M]
[2M]
[4M]
Hence the general solution is
[2M]
(b) Show that every non trivial solution of
has infinitely
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
Q 1. A)
2
2nd order & non linear equation since
x is missing. Let
&
are of 2nd degree
[4 M]
=p
.
; p=0 cant be considered
.
const.
[1 M]
. I.F. =
2/3
[5 M]
[5 M]
Q 1 B)
My = x ; Nx= 4x => My Nx
[6 M]
[4 M]
[2 M]
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
Q2 (a)
As
, dividing by
we get the equation
Where
(1 pt)
This equation can be converted to an equation with constant coefficients if and only if
for some constant c
(2 pts)
2 x 2 x 2 P( x )
c
1
c P( x ) x
3
x
2
x
c
( x ) x 2 1.
2
(1) 1 c 2. ( x) x 2 1
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
First Semester 200910 MATH C241 Mathematics III
Comprehensive Examination (Closed Book)
Date: 01052010
Max. Marks: 80
1. Solve
log
2
0.
[8]
2. Find particular solution of the following solution using th
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
Birla Institute of Technology and Science, Pilani
I semester 20072008
Math C241 : Mathematics III
Comprehensive Examination (closed book)
Date: 03/12/2007
Duration : 3 hours
MM : 120
NOTE: 1. Answer Part A and Part B in separate answer books provided.
2
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
SET A
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI
I Semester 20072008, MATH C 241 MATHEMATICS III
QUIZ (Closed Book)
Duration: 50 mins
MM: 60
Name:
Sec No. :
Id. No. :
Instructors Name:
Cutting or overwriting will be treated as not attempted. Write
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
II SEMESTER 20052006
MATH C241: MathematicsIII
Comprehensive Examination
Date: May 05, 2006
Day: Friday
Time: 3 hrs.
Max. Marks: 40
Q1. Find the general solution of the homogeneous linear system:
dx
dy
=
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
First Semester 200607 MATH C241 Mathematics III
Comprehensive Examination (Closed Book) Part B & C
Date: 71206
Max. Marks: 90
Note: (1) The paper consists of two parts: Part B and Part C.
(2) Both parts
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Winter 2014
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
I SEMESTER 20052006
MATH C241: MathematicsIII
Comprehensive Examination
Date: December 02, 2005
hrs.
Day: Friday
Time: 3
Max. Marks: 120
Note: Answer Part A and Part B on separate answer books and write
Birla Institute of Technology & Science, Pilani  Hyderabad
Differential Equations
MATHEMATIC 211

Spring 2014
Differential Equations
MATH C241
Class hours: T Th S 2
(9.00 A.M. to 9.50 A.M.)
Text Book:
Differential Equations
with Applications and
Historical Notes:
by George F. Simmons
(Tata McGrawHill) (2003)
In this introductory lecture, we
Define a differential