MA 201: Lecture - 9
One dimensional wave equation
MA201(2017):PDE
The DAlemberts solution of the wave equation
Consider the one-dimensional wave equation:
utt = c 2 uxx , x R, t > 0, c R is a constant.
Here as B 2 4AC = 4c 2 > 0, (1) is a hyperbolic PDE.
MA 201: Fourier Series
Lecture - 10
MA201(2017):PDE
HISTORICAL REMARKS on Fourier Series
The theory of Fourier series had its historical origin in the middle of
the eighteenth century, when several mathematicians were studying
the vibrations of stretched
End Semester Examination: MA 201 Mathematics III
Department of Mathematics, IIT Guwahati
Date: November 25, 2016
Time: 9:0012:00 hours
Name:
Maximum Marks: 50
Roll No:
Instructions:
(a) Write your name and roll number above as soon as you receive the ques
Solutions of Quiz 2: MA 201 Mathematics III
Department of Mathematics, IIT Guwahati
Date: November 1, 2016
Time: 8:00-8:45 AM
Total Marks: 10
1. Find the partial dierential equation of all spheres of xed radius having
their centres in the xy-plane.
[2]
So
End Semester Examination: MA 201 Mathematics III
Department of Mathematics, IIT Guwahati
Date: November 25, 2016
Time: 9:0012:00 hours
Name:
Maximum Marks: 50
Roll No:
Instructions:
(a) Write your name and roll number above as soon as you receive the ques
MA 201
Partial Differential Equations
Tutorial Problems - 1
Formulation and classification of PDEs, Method of characteristics, Cauchy problems
1. Find the partial differential equation arising from each of the following surfaces and
classify them as linea
MA 201
Partial Differential Equations
Tutorial Problems - 2
Topics: Orthogonal surface, Solution of 1st order nonlinear PDEs by method of characteristics
Complete integral of 1st order nonlinear PDEs
1. Find the surface which is orthogonal to the one-para
Quiz 2: MA 201 Mathematics III
Department of Mathematics, IIT Guwahati
Date: November 1, 2016
Time: 8:00-8:45 AM
Total Marks: 10
1. Find the partial dierential equation of all spheres of xed radius having
their centres in the xy-plane.
[2]
2. A river is d
MA 201
Partial Differential Equations
Answers-Tutorial Sheet-2
1. a4 (x2 + y 2 ) = (x2 y 2 )2 (x2 + y 2 + 4u2 ).
2
2
2. F (x2 + y 2 + u2 , x uy
).
2
3. u = x + c(1 + xy), where c is a parameter.
p
4. (i) 8u = (2 2
1 + x2+ y 2 )2.
2
2y
1
x2
(ii) u(x, y) =
MA 201: Lecture - 8
Second order PDEs: Canonical forms (cont.)
MA201(2017):PDE
Canonical form for parabolic equation: D = 0.
General second-order semi-linear partial differential equation in two
independent variables x, t is of the form
Auxx + Buxt + Cutt
MA 201
Partial Differential Equations
Tutorial Sheet-3
Jacobi method for I order PDEs, Classification of 2nd order PDEs, Canonical/Normal forms.
1. Let u = u(x1 , x2 , x3 ) and ui := uxi , where x1 , x2 , x3 are independent variables. Using Jacobi
method
MA 102
Mathematics II
Lecture 7
13 March, 2015
Linear higher-order DEs-IVPs
For a linear differential equation, an n-th order initial-value problem is
Solve : an (x)
dn1 y
dy
dn y
+ an1 (x) n1 + + a1 (x)
+ a0 (x)y = g(x)
n
dx
dx
dx
subject to : y(x0 ) = y
Lecture 11, MA 102
25 March, 2015
Variation of parameters
Let us first see a procedure (called VARIATION OF PARAMETERS
procedure) to find a particular solution of a linear first-order DE
dy
+ P (x)y = f (x)
dx
(1)
on an interval I om which the functions P
Multivariable Calculus
Lecture 5 on 13.01.2014 (Monday)
(MA102 Mathematics II)
M. G. P. Prasad
IIT Guwahati
M. G. P. Prasad ( IIT Guwahati )
Multivariable Calculus Lecture 5 on 13.01.2014 (Monday)
1 / 27
Learning Outcome of this Lecture
We learn
Continuit
Lecture 12, MA 102
26 March, 2015
Systems of Linear Equations (with constant
coefficients)
d
Let t denote an independent variable, D denote the differential operator dt
,
and m and n be integers 2. Let x1 , x2 , . . . , xn denote n many variables
which de
Lecture 8, MA 102
18 March, 2015
Recall
Recall the homogeneous linear n-th order DE
an (x)
dn1 y
dy
dn y
+ an1 (x) n1 + + a1 (x)
+ a0 (x)y = 0
n
dx
dx
dx
()
Fundamental set of solutions
Definition
Any set cfw_y1 , y2 , . . . , yn of n linearly independen
MA 102
Mathematics II
Lecture 3
4 March, 2015
Initial value problem (IVP)
A problem to the type
On some interval I containing x0
Find a solution y(x) of the ODE
F (x, y, y 0 , . . . , y (n) ) = 0
subject to the conditions
y(x0 ) = a0 , y 0 (x0 ) = a1 , .
MA 102
Mathematics II
Lecture 2
3 March, 2015
Interval of Definition
Recall :
Definition
A function defined on an interval I and possessing at least n derivatives,
which when substituted into an n-th order ODE reduces the equation to an
identity, is said
DEPARTMENT OF MATHEMATICS, IIT - GUWAHATI
Even Semester of the Academic year 2014 - 2015
MA 102 Mathematics II
Problem Sheet 4: Method of undetermined coefficients, annihilator approach
(operator method), variation of parameters.
Instructor: Dr. J. C. Kal
Indian Institute of Technology Guwahati
Solution to Tutorial Sheet 5
Jan-April 2015 semester
MA 102, Mathematics II
n
Ans 1: (a) The given power series is of the form
n=1 an (x0) where an =
limn |
2n
n .
an+1
n2n+1
|x| = limn |
|x| = 2|x|.
an
(n + 1)2n
T
Lecture 14, MA 102
1 April, 2015
Recall
What is meant by saying that A DIFF EQN () HAS A POWER SERIES
SOLUTION IN (x a) OR CENTERED at a?: We mean that there exists
coefficients cn and an open interval I containing the point a such that the
n
power series
Indian Institute of Technology Guwahati
Partial solution to Tutorial Sheet 4
Jan-April 2015 semester
MA 102, Mathematics II
1. Solve the given differential equations by the method of undetermined coefficients.
(a) y 00 8y 0 + 20y = 100x2 26xex .
(b) y 00
Indian Institute of Technology Guwahati
Solution to Tutorial Sheet 4
Jan-April 2015 semester
MA 102, Mathematics II
Ans 1: (a) The given DE can be rewritten as y 00 8y 0 + 20y = g(x), where
g(x) = 100x2 26xex . The associated homogeneous equation is y 00
MA 102
Mathematics II
Lecture 4
5 March, 2015
First order ODE s
We will now discuss different methods of solutions of first order ODEs. The
first type of such ODEs that we will consider is the following:
Definition
Separable variables: A first order diffe
Lecture 13, MA 102
27 March, 2015
Review of Power series
Most linear differential equations with variable coefficients cannot be solved
in terms of elementary functions. A standard technique for solving
higher-order linear differential equations with vari
Lecture 15 & partial of Lecture 16, MA 102
8 & 9 April, 2015
Recall: Singular points
Consider a general linear 2 nd order DE:
a2 (x)y 00 + a1 (x)y 0 + a0 (x)y = 0.
(1)
Putting this equation in standard form, we get
y 00 + P (x)y 0 + Q(x)y = 0.
(2)
If x0 i
Lecture 10, MA 102
20 March, 2015
Case I
Recall the equation
an y (n) + an1 y (n1) + + a1 y 0 + a0 y = g(x)
(1)
Case I: No function in the assumed particular solution is a solution of the
associated homogeneous DE.
In the table that follows, we illustrate
Lecture 21, MA 102
23 April, 2015
Stability Analysis
Consider the plane autonomous system
X 0 = AX
(1)
x
is the matrix having constant entries and X =
.
y
0
To ensure that X0 =
is the only critical point, we will assume that the
0
determinant of A given b