MA 102, PART II (DIFFERENTIAL EQUATIONS), TUTORIAL 1
SEMESTER II, AY 20152016
Problem 1 Find the general solution of the following ode:
(1) y 0 = x2 (1 + y 2 )
(2) y 0 = x2 /(y + x3 y)
(3) y 0 = (4x2 + y 2 )/(xy)
(4) 1 + y 2 + y 2 y 0 = 0
(5) y 0 + (1/x)y

MA 102, PART II (DIFFERENTIAL EQUATIONS),
TUTORIAL 4
SEMESTER II, AY 20152016
Problem 1 Solve the Euler equations
(1) x2 y 00 + xy 0 + y = 0
(2) x2 y 00 4xy 0 6y = 0
Problem 2 Use the method of reduction of order to find a second solution
of the given dif

MA 102, PART II (DIFFERENTIAL EQUATIONS), TUTORIAL 3
SEMESTER II, AY 20152016
Problem 1 The amount x(t) of a radioactive substance at time t is governed by
dx
= kx where k > 0. If the half-life of such a radioactive
an ode of the form
dt
substance is 20 d

MA 102, PART II (DIFFERENTIAL EQUATIONS), TUTORIAL 3
SEMESTER II, AY 20152016
Problem 1 The amount x(t) of a radioactive substance at time t is governed by
dx
= kx where k > 0. If the half-life of such a radioactive
an ode of the form
dt
substance is 20 d

MA 102, PART II (DIFFERENTIAL EQUATIONS), TUTORIAL 2
SEMESTER II, AY 20152016
Problem 1 Determine whether or not each of the following ode is exact. If it is
exact, find the solution.
i. 2x + 4y + (2x 2y)y 0 = 0.
ii. 2xy 2 + 2y + (2x2 y + 2x)y 0 = 0.
iii.

Maxima and minima
Maxima and Minima
Department of Mathematics
IIT Guwahati
RA/RKS/PASS
IITG: MA-102 (2016)
Maxima and minima
Local extremum of f : Rn R
Let f : U Rn R be continuous, where U is open. Then
f has a local maximum at p if there exists r > 0 su

Constrained extrema and Lagrange multipliers
Constrained extrema and Lagrange multipliers
Inverse and implicit function theorems
Department of Mathematics
IIT Guwahati
RA/RKS/PASS
IITG: MA-102 (2016)
Constrained extrema and Lagrange multipliers
Constraine

Continuous functions
Lecture Slide 2:
Continuous functions
Department of Mathematics
IIT Guwahati
RA/RKS/PASS
MA-102 (2016)
Continuous functions
Continuous functions
Task: Analyze continuity of the functions:
Case I: f : A Rn R
Case II: f : A R Rn
Case II

Partial and Directional Derivatives
Lecture Slides 4:
Partial and Directional derivatives
Department of Mathematics
IIT Guwahati
RA/RKS/PASS
MA-102 (2016)
Partial and Directional Derivatives
Differential Calculus
Task: Extend differential calculus to the

Vector Fields, Curl and Divergence
Vector fields, Curl and Divergence
Department of Mathematics
IIT Guwahati
RA/RKS/PASS
IITG: MA-102 (2016)
Vector Fields, Curl and Divergence
Vector fields
Definition: A vector field in Rn is a function F : Rn Rn that
ass

Limit and continuity of functions
Lecture Slides 3:
Limit and Continuity of Functions
Department of Mathematics
IIT Guwahati
RA/RKS/PASS
MA-102 (2016)
Limit and continuity of functions
Topology of Rn
Open Ball: Let > 0 and a Rn . Then
B(a, ) := cfw_x Rn :

Arclength and Line Integrals
Arclength and Line Integrals
Department of Mathematics
IIT Guwahati
RA/RKS/PASS
IITG: MA-102 (2016)
Arclength and Line Integrals
Parametric curves
Definition:
A continuous mapping : [a, b] Rn is called a
parametric curve or a

Chain rule
Tangents and Normals
Higher order derivatives
Chain rule, Tangents and Higher order
derivatives
Department of Mathematics
IIT Guwahati
RA/RKS/PASS
IITG: MA-102 (2016)
Chain rule
Tangents and Normals
Higher order derivatives
Chain rule
Theorem-A

Differentiability
Lecture Slides 5:
Differentiability of functions of several variables
Department of Mathematics
IIT Guwahati
RA/RKS/PASS
IITG: MA-102 (2016)
Differentiability
Differential Calculus for f : Rn R
Question: Let f : Rn R. What does it mean t

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