(UNIVERSITY OF MUMBAI)
Syllabus for: S.Y.B.Sc./S.Y.B.A.
Program: B.Sc./B/A.
Course: Mathematics
Choice based Credit System (CBCS)
with effect from the
academic year 2018-19
2SEMESTER III
CALCULUS III
Course Code UNIT TOPICS Credits L/Week
USMT 301, UAMT 301
I Functions of several variables
2 3II Differentiation
III Applications
ALGEBRA III
USMT 302 ,UAMT 302
I Linear Transformations and Matrices
2 3II Determinants
III Inner Product Spaces
DISCRETE MATHEMATICS
USMT 303
I Permutations and Recurrence Relation
2 3II Preliminary Counting
III Advanced Counting
PRACTICALS
USMTP03
Practicals based on
3 5
USMT301, USMT 302 and USMT 303
UAMTP03
Practicals based on
2 4
UAMT301, UAMT 302
SEMESTER IV
CALCULUS IV
Course Code UNIT TOPICS Credits L/Week
USMT 401, UAMT 401
I Riemann Integration
2 3
II Indefinite Integrals and Improper Integrals
III Beta and Gamma Functions
And Applications
ALGEBRA IV
USMT 402 ,UAMT 402
I Groups and Subgroups
2 3II Cyclic Groups and Cyclic subgroups
III Lagrange’s Theorem and Group
Homomorphism
ORDINARY DIFFERENTIAL EQUATIONS
USMT 403
I First order First degree
2 3Differential equations
II Second order Linear
Differential equations
III Linear System of Ordinary
Differential Equations
PRACTICALS
USMTP04
Practicals based on
3 5
USMT401, USMT 402 and USMT 403
UAMTP04
Practicals based on
2 4
UAMT401, UAMT 402
3Teaching Pattern for Semester III
1. Three lectures per week per course. Each lecture is of 48 minutes duration.
2. One Practical (2L) per week per batch for courses USMT301, USMT 302 combined and
one Practical (3L) per week for course USMT303 (the batches tobe formed as prescribed
by the University. Each practical session is of 48 minutes duration.)
Teaching Pattern for Semester IV
1. Three lectures per week per course. Each lecture is of 48 minutes duration.
2. One Practical (2L) per week per batch for courses USMT301, USMT 302 combined and
one Practical (3L) per week for course USMT303 (the batches tobe formed as prescribed
by the University. Each practical session is of 48 minutes duration.)
S.Y.B.Sc. / S.Y.B.A. Mathematics
SEMESTER III
USMT 301, UAMT 301: CALCULUS III
Note: All topics have to be covered with proof in details (unless mentioned otherwise) and
examples.
Unit I: Functions of several variables (15 Lectures)
1. The Euclidean inner product on Rn and Euclidean norm function on Rn, distance between
two points, open ball in Rn, definition of an open subset of Rn, neighbourhood of a point
in Rn, sequences in Rn, convergence of sequences- these concepts should be specifically
discussed for n = 3 and n = 3.
2. Functions from Rn −→ R (scalar fields) and from Rn −→ Rm (vector fields), limits,
continuity of functions, basic results on limits and continuity of sum, difference, scalar
multiples of vector fields, continuity and components of a vector fields.
3. Directional derivatives and partial derivatives of scalar fields.
4. Mean value theorem for derivatives of scalar fields.
Reference for Unit I:
Sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 8.10 of Calculus, Vol. 2 (Second Edition) by
Apostol.
Unit II: Differentiation (15 Lectures)
1. Differentiability of a scalar field at a point of Rn (in terms of linear transformation) and
on an open subset of Rn, the total derivative, uniqueness of total derivative of a differ-
entiable function at a point, simple examples of finding total derivative of functions such
as f(x, y) = x2 + y2, f(x, y, z) = x + y + z,, differentiability at a point of a function f
implies continuity and existence of direction derivatives of f at the point, the existence of
continuous partial derivatives in a neighbourhood of a point implies differentiability at the
point.
42. Gradient of a scalar field, geometric properties of gradient, level sets and tangent planes.
3. Chain rule for scalar fields.
4. Higher order partial derivatives, mixed partial derivatives, sufficient condition for equality
of mixed partial derivative.
Reference for Unit II:
Sections 8.11, 8.12, 8.13, 8.14, 8.15, 8.16, 8.17, 8.23 of Calculus, Vol.2 (Second Edition) by T.
Apostol, John Wiley.
Unit III: Applications (15 lectures)
1. Second order Taylor’s formula for scalar fields.
2. Differentiability of vector fields, definition of differentiability of a vector field at a point,
Jacobian matrix, differentiability of a vector field at a point implies continuity. The chain
rule for derivative of vector fields (statements only)
3. Mean value inequality.
4. Hessian matrix, Maxima, minima and saddle points.
5. Second derivative test for extrema of functions of two variables.
6. Method of Lagrange Multipliers.
Reference for Unit III:
Sections 8.18, 8.19, 8.20, 8.21, 8.22, 9.9, 9.10, 9.11, 9.12, 9.13, 9.14 9.13, 9.14 from Apostol,
Calculus Vol. 2, (Second Edition) by T. Apostol.
Recommended Text Books:
1. T. Apostol: Calculus, Vol. 2, John Wiley.
2. J. Stewart, Calculus, Brooke/ Cole Publishing Co.
Additional Reference Books
(1) G.B. Thoman and R. L. Finney, Calculus and Analytic Geometry, Ninth Edition, Addison-
Wesley, 1998.
(2) Sudhir R. Ghorpade and Balmohan V. Limaye, A Course in Multivariable Calculus and
Analysis, Springer International Edition.
(3) Howard Anton, Calculus- A new Horizon, Sixth Edition, John Wiley and Sons Inc, 1999.
USMT 302/UAMT 302: ALGEBRA III
Note: Revision of relevant concepts is necessary.
Unit 1: Linear Transformations and Matrices (15 lectures)
51. Review of linear transformations: Kernel and image of a linear transformation, Rank-
Nullity theorem (with proof), Linear isomorphisms, inverse of a linear isomorphism, Any
n− dimensional real vector space is isomorphic to Rn.
2. The matrix units, row operations, elementary matrices, elementary matrices are invertible
and an invertible matrix is a product of elementary matrices.
3. Row space, column space of an m × n matrix, row rank and column rank of a matrix,
Equivalence of the row and the column rank, Invariance of rank upon elementary row or
column operations.
4. Equivalence of rank of an m × n matrix A and rank of the linear transformation LA :
Rn −→ Rm (LA(X) = AX). The dimension of solution space of the system of linear
equations AX = 0 equals n−rank(A).
5. The solutions of non-homogeneous systems of linear equations represented by AX = B,
Existence of a solution when rank(A)= rank(A,B), The general solution of the system is
the sum of a particular solution of the system and the solution of the associated homoge-
neous system.
Reference for Unit 1: Chapter VIII, Sections 1, 2 of Introduction to Linear Algebra,
Serge Lang, Springer Verlag and Chapter 4, of Linear Algebra A Geometric Approach, S.
Kumaresan, Prentice-Hall of India Private Limited, New Delhi.
Unit II: Determinants (15 Lectures)
1. Definition of determinant as an n−linear skew-symmetric function from Rn × Rn × . . . ×
Rn −→ R such that determinant of (E1, E2, . . . , En) is 1, where Ej denotes the jth column
of the n × n identity matrix In. Determinant of a matrix as determinant of its column
vectors (or row vectors). Determinant as area and volume.
2. Existence and uniqueness of determinant function via permutations, Computation of de-
terminant of 2 × 2, 3 × 3 matrices, diagonal matrices, Basic results on determinants such
as det(At) = det(A), det(AB) = det(A) det(B), Laplace expansion of a determinant, Van-
dermonde determinant, determinant of upper triangular and lower triangular matrices.
3. Linear dependence and independence of vectors in Rn using determinants, The existence
and uniqueness of the system AX = B, where A is an n×n matrix wither det(A) 6= 0, Co-
factors and minors, Adjoint of an n×n matrix A, Basic results such as Aadj(A) = det(A)In.
An n× n real matrix A is invertible if and only if det(A) 6= 0, A−1 = 1
det(A)
adj(A) for an
invertible matrix A, Cramer’s rule.
4. Determinant as area and volume.
References for Unit 2: Chapter VI of Linear Algebra A geometric approach, S. Kumaresan,
Prentice Hall of India Private Limited, 2001 and Chapter VII Introduction to Linear Algebra,
Serge Lang, Springer Verlag.
Unit III: Inner Product Spaces (15 Lectures)
1. Dot product in Rn, Definition of general inner product on a vector space over R. Examples
of inner product including the inner product < f, g >=
∫ pi
−pi
f(t)g(t) dt on C[−pi, pi], the
space of continuous real valued functions on [−pi, pi].
62. Norm of a vector in an inner product space. Cauchy-Schwartz inequality, Triangle in-
equality, Orthogonality of vectors, Pythagoras theorem and geometric applications in R2,
Projections on a line, The projection being the closest approximation, Orthogonal com-
plements of a subspace, Orthogonal complements in R2 and R3. Orthogonal sets and
orthonormal sets in an inner product space, Orthogonal and orthonormal bases. Gram-
Schmidt orthogonalization process, Simple examples in R3,R4.
Reference of Unit 3: Chapter VI, Sections 1,2 of Introduction to Linear Algebra, Serge
Lang, Springer Verlag and Chapter 5, of Linear Algebra A Geometric Approach, S. Kumaresan,
Prentice-Hall of India Private Limited, New Delhi.
Recommended Books:
1. Serge Lang: Introduction to Linear Algebra, Springer Verlag.
2. S. Kumaresan: Linear Algebra A geometric approach, Prentice Hall of India Private Lim-
ited.
Additional Reference Books:
1. M. Artin: Algebra, Prentice Hall of India Private Limited.
2. K. Hoffman and R. Kunze: Linear Algebra, Tata McGraw-Hill, New Delhi.
3. Gilbert Strang: Linear Algebra and its applications, International Student Edition.
4. L. Smith: Linear Algebra, Springer Verlag.
5. A. Ramachandra Rao and P. Bhima Sankaran: Linear Algebra, Tata McGraw-Hill, New
Delhi.
6. T. Banchoff and J. Wermer: Linear Algebra through Geometry, Springer Verlag Newyork,
1984.
7. Sheldon Axler: Linear Algebra done right, Springer Verlag, Newyork.
8. Klaus Janich: Linear Algebra.
9. Otto Bretcher: Linear Algebra with Applications, Pearson Education.
10. Gareth Williams: Linear Algebra with Applications, Narosa Publication.
USMT 303: Discrete Mathematics
Unit I: Permutations and Recurrence relation (15 lectures)
1. Permutation of objects, Sn, composition of permutations, results such as every permutation
is a product of disjoint cycles, every cycle is a product of transpositions, even and odd
permutation, rank and signature of a permutation, cardinality of Sn, An
2. Recurrence Relations, definition of non-homogeneous, non-homogeneous, linear , non-
linear recurrence relation, obtaining recurrence relation in counting problems, solving
homogeneous as well as non homogeneous recurrence relations by using iterative meth-
ods, solving a homogeneous recurrence relation of second degree using algebraic method
proving the necessary result.
7Recommended Books:
1. Norman Biggs: Discrete Mathematics, Oxford University Press.
2. Richard Brualdi: Introductory Combinatorics, John Wiley and sons.
3. V. Krishnamurthy: Combinatorics-Theory and Applications, Affiliated East West Press.
4. Discrete Mathematics and its Applications, Tata McGraw Hills.
5. Schaum’s outline series: Discrete mathematics,
6. Applied Combinatorics: Allen Tucker, John Wiley and Sons.
Unit II: Preliminary Counting (15 Lectures)
1. Finite and infinite sets, countable and uncountable sets examples such as N,Z,N× N,Q,(0, 1),R
2. Addition and multiplication Principle, counting sets of pairs, two ways counting.
3. Stirling numbers of second kind. Simple recursion formulae satisfied by S(n, k) for k =
1, 2, · · · , n− 1, n
4. Pigeonhole principle and its strong form, its applications to geometry, monotonic sequences
etc.
Unit III: Advanced Counting (15 Lectures)
1. Binomial and Multinomial Theorem, Pascal identity, examples of standard identities such
as the following with emphasis on combinatorial proofs.
ˆ
r∑
k=0
(
m
k
)(
n
r − k
)
=
(
m+ n
r
)
ˆ
n∑
i=r
(
i
r
)
=
(
n+ 1
r + 1
)
ˆ
k∑
i=0
(
k
i
)2
=
(
2k
k
)
ˆ
n∑
i=0
(
n
i
)
= 2n
2. Permutation and combination of sets and multi-sets, circular permutations, emphasis on
solving problems.
3. Non-negative and positive solutions of equation x1 + x2 + · · ·+ xk = n
4. Principal of inclusion and exclusion, its applications, derangements, explicit formula for
dn, deriving formula for Euler’s function φ(n).
8USMT P03/UAMTP03 Practicals
Suggested Practicals for USMT 301/UAMT303
1. Sequences in R2 and R3, limits and continuity of scalar fields and vector fields, using
“definition and otherwise” , iterated limits.
2. Computing directional derivatives, partial derivatives and mean value theorem of scalar
fields.
3. Total derivative, gradient, level sets and tangent planes.
4. Chain rule, higher order derivatives and mixed partial derivatives of scalar fields.
5. Taylor’s formula, differentiation of a vector field at a point, finding Hessian/Jacobean
matrix, Mean Value Inequality.
6. Finding maxima, minima and saddle points, second derivative test for extrema of functions
of two variables and method of Lagrange multipliers.
7. Miscellaneous Theoretical Questions based on full paper
Suggested Practicals for USMT302/UAMT302:
1. Rank-Nullity Theorem.
2. System of linear equations.
3. Determinants , calculating determinants of 2×2 matrices, n×n diagonal, upper triangular
matrices using definition and Laplace expansion.
4. Finding inverses of n× n matrices using adjoint.
5. Inner product spaces, examples. Orthogonal complements in R2 and R3.
6. Gram-Schmidt method.
7. Miscellaneous Theoretical Questions based on full paper
Suggested Practicals for USMT 303:
1. Derangement and rank signature of permutation.
2. Recurrence relation.
3. Problems based on counting principles, Two way counting.
4. Stirling numbers of second kind, Pigeon hole principle.
5. Multinomial theorem, identities, permutation and combination of multi-set.
6. Inclusion-Exclusion principle. Euler phi function.
7. Miscellaneous theory quesitons from all units.
9SEMESTER IV
USMT 401/UAMT 401: CALCULUS IV
Note: All topics have to be covered with proof in details (unless mentioned otherwise) and
examples.
Unit I: Riemann Integration (15 Lectures)
Approximation of area, Upper/Lower Riemann sums and properties, Upper/Lower integrals,
Definition of Riemann integral on a closed and bounded interval, Criterion of Riemann in-
tegrability, if a < c < b then f ∈ R[a, b], if and only if f ∈ R[a, c] and f ∈ R[c, b] and∫ b
a
f =
∫ c
a
f +
∫ b
c
f .
Properties:
(i) f, g ∈ R[a, b] =⇒ f + g , λf ∈ R[a, b].
(ii)
∫ b
a
(f + g) =
∫ b
a
f +
∫ b
a
g.
(iii)
∫ b
a
λf = λ
∫ b
a
f.
(iv) f ∈ R[a, b] =⇒ |f | ∈ R[a, b] and |
∫ b
a
f | ≤
∫ b
a
|f |,
(v) f ≥ 0, f ∈ C [a, b] =⇒ f ∈ R[a, b].
(vi) If f is bounded with finite number of discontinuities then f ∈ R[a, b], generalize this if f
is monotone then f ∈ R[a, b].
Unit II: Indefinite and improper integrals (15 lectures)
Continuity of F (x) =
∫ x
a
f(t) dt where f ∈ R[a, b], Fundamental theorem of calculus, Mean
value theorem, Integration by parts, Leibnitz rule, Improper integrals-type 1 and type 2, Abso-
lute convergence of improper integrals, Comparison tests, Abel’s and Dirichlet’s tests.
Unit III: Applications (15 lectures)
(1) β and Γ functions and their properties, relationship between β and Γ functions (without
proff).
(2) Applications of definite Integras: Area between curves, finding volumes by sicing, volumes
of solids of revolution-Disks and Washers, Cylindrical Shells, Lengths of plane curves, Ar-
eas of surfaces of revolution.
References:
(1) Calculus Thomas Finney, ninth edition section 5.1, 5.2, 5.3, 5.4, 5.5, 5.6.
(2) R. R. Goldberg, Methods of Real Analysis, Oxford and IBH, 1964.
10
(3) Ajit Kumar, S.Kumaresan, A Basic Course in Real Analysis, CRC Press, 2014.
(4) T. Apostol, Calculus Vol.2, John Wiley.
(5) K. Stewart, Calculus, Booke/Cole Publishing Co, 1994.
(6) J. E. Marsden, A.J. Tromba and A. Weinstein, Basic multivariable calculus.
(7) Bartle and Sherbet, Real analysis.
USMT 402/ UAMT 402: ALGEBRA IV
Unit I: Groups and Subgroups (15 Lectures)
(a) Definition of a group, abelian group, order of a group, finite and infinite groups.
Examples of groups including:
i) Z,Q,R,C under addition.
ii) Q∗(= Q \ {0}),R∗(= R \ {0}),C∗(= C \ {0}).Q+(= positive rational numbers)
under multiplication.
iii) Zn, the set of residue classes modulo n under addition.
iv) U(n), the group of prime residue classes modulo n under multiplication.
v) The symmetric group Sn.
vi) The group of symmetries of a plane figure. The Dihedral group Dn as the group
of symmetries of a regular polygon of n sides (for n = 3, 4).
vii) Klein 4-group.
viii) Matrix groups Mn×n(R) under addition of matrices, GLn(R), the set of invertible
real matrices, under multiplication of matrices.
ix) Examples such as S1 as subgroup of C , µn the subgroup of n−th roots of unity.
(b) Properties such as
1) In a group (G, .) the following indices rules are true for all integers n,m.
i) anam = an+m for all a in G.
ii) (an)m = anm for all a in G.
iii) (ab)n = anbn for all ab in G whenever ab = ba.
2) In a group (G, .) the following are true:
i) The identity element e of G is unique.
ii) The inverse of every element in G is unique.
iii) (a−1)−1 = a for all a in G.
iv) (a.b)−1 = b−1a−1 for all a, b in G.
v) If a2 = e for every a in G then (G, .) is an abelian group.
vi) (aba−1)n = abna−1 for every a, b in G and for every integer n.
vii) If (a.b)2 = a2.b2 for every a, b in G then (G, .) is an abelian group.
viii) (Z∗n, .) is a group if and only if n is a prime.
3) Properties of order of an element such as: ( n and m are integers.)
i) If o(a) = n then am = e if and only if n/m.
ii) If o(a) = nm then o(an) = m.
iii) If o(a) = n then o(am) =
n
(n,m)
,. where (n,m) is the GCD of n and m.
11
iv) o(aba−1) = o(b) and o(ab) = o(ba).
v) If o(a) = m and o(b) = m, ab = ba, (n,m) = 1 then o(ab) = nm.
(c) Subgroups
i) Definition, necessary and sufficient condition for a non-empty set to be a Sub-
group.
ii) The center Z(G) of a group is a subgroup.
iii) Intersection of two (or a family of ) subgroups is a subgroup.
iv) Union of two subgroups is not a subgroup in general. Union of two subgroups is
a subgroup if and only if one is contained in the other.
v) If H and K are subgroups of a group G then HK is a subgroup of G if and only
if HK = KH.
Reference for Unit I:
(1) I.N. Herstein, Topics in Algebra.
(2) P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra.
Unit II: Cyclic groups and cyclic subgroups (15 Lectures)
(a) Cyclic subgroup of a group, cyclic groups, (examples including Z,Zn and µn).
(b) Properties such as:
(i) Every cyclic group is abelian.
(ii) Finite cyclic groups, infinite cyclic groups and their generators.
(iii) A finite cyclic group has a unique subgroup for each divisor of the order of the group.
(iv) Subgroup of a cyclic group is cyclic.
(v) In a finite group G,G =< a > if and only if o(G) = o(a).
(vi) If G =< a > and o(a) = n then G =< am > if and only if (n,m) = 1.
(vii) If G is a cyclic group of order pn and H < G,K < G then prove that either H ⊆ K
or K ⊆ H.
References for Unit II:
(1) I.N. Herstein, Topics in Algebra.
(2) P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra.
Unit III: Lagrange’s Theorem and Group homomorphism (15 Lectures)
(a) Definition of Coset and properties such as :
1) IF H is a subgroup of a group G and x ∈ G then
(i) xH = H if and only if x ∈ H.
(ii) Hx = H if and only if x ∈ H.
2) If H is a subgroup of a group G and x, y ∈ G then
(i) xH = yH if and only if x−1y ∈ H.
(ii) Hx = Hy if and only if xy−1 ∈ H.
12
3) Lagrange’s theorem and consequences such as Fermat’s Little theorem, Euler’s theo-
rem and if a group G has no nontrivial subgroups then order of G is a prime and G
is Cyclic.
(b) Group homomorphisms and isomorphisms, automorphisms
i) Definition.
ii) Kernel and image of a group homomorphism.
iii) Examples including inner automorphism.
Properties such as:
(1) f : G −→ G′ is a group homomorphism then ker f < G.
(2) f : G −→ G′ is a group homomorphism then ker f = {e} if and only if f is 1-1.
(3) f : G −→ G′ is a group homomorphism then
(i) G is abelian if and only if G′ is abelian.
(ii) G is cyclic if and only if G′ is cyclic.
Reference for Unit III:
1. I.N. Herstein, Topics in Algebra.
2. P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra.
Recommended Books:
1. I.N. Herstein, Topics in Algebra, Wiley Eastern Limied, Second edition.
2. N.S. Gopalkrishnan, University Algebra, Wiley Eastern Limited.
3. M. Artin, Algebra, Prentice Hall of India, New Delhi.
4. P.B. Bhattacharya, S.K. Jain, S. Nagpaul. Abstract Algebra, Second edition, Foundation
Books, New Delhi, 1995.
5. J.B. Fraleigh, A first course in Abstract Algebra, Third edition, Narosa, New Delhi.
6. J. Gallian. Contemporary Abstract Algebra. Narosa, New Delhi.
7. COmbinatroial Techniques by Sharad S. Sane, Hindustan Book Agency.
Additional Reference Books:
1. S. Adhikari. An introduction to Commutative Algebra and Number theory. Narosa Pub-
lishing House.
2. T. W. Hungerford. Algebra, Springer.
3. D. Dummit, R. Foote. Abstract Algebra, John Wiley & Sons, Inc.
4. I.S. Luther, I.B.S. Passi. Algebra. Vol. I and II.
13
USMT 403: ORDINARY DIFFERENTIAL EQUATIONS
Unit I: First order First degree Differential equations (15 Lectures)
(1) Definition of a differential equation, order, degree, ordinary differential equation and partial
differential equation, linear and non linear ODE.
(2) Existence and Uniqueness Theorem for the solution of a second order initial value prob-
lem (statement only), Definition of Lipschitz function, Examples based on verifying the
conditions of existence and uniqueness theorem
(3) Review of Solution of homogeneous and non-homogeneous differential equations of first
order and first degree. Notion of partial derivatives. Exact Equations: General solution
of Exact equations of first order and first degree. Necessary and sufficient condition for
Mdx + Ndy = 0 to be exact. Non-exact equations: Rules for finding integrating factors
(without proof) for non exact equations, such as :
i)
1
M x+N y
is an I.F. if M x+N y 6= 0 and Mdx+Ndy = 0 is homogeneous.
ii)
1
M x−N y is an I.F. if M x − N y 6= 0 and Mdx + Ndy = 0 is of the form
f1(x, y) y dx+ f2(x, y) x dy = 0.
iii) e
∫
f(x) dx (resp e
∫
g(y) dy ) is an I.F. if N 6= 0 (resp M 6= 0) and 1
N
(
∂M
∂y
− ∂N
∂x
)
(
resp
1
M
(
∂M
∂y
− ∂N
∂x
))
is a function of x (resp y) alone, say f(x) (resp g(y)).
iv) Linear and reducible linear equations of first order, finding solutions of first order dif-
ferential equations of the type for applications to orthogonal trajectories, population
growth, and finding the current at a given time.
Unit II: Second order Linear Differential equations (15 Lectures)
1. Homogeneous and non-homogeneous second order linear differentiable equations: The
space of solutions of the homogeneous equation as a vector space. Wronskian and linear
independence of the solutions. The general solution of homogeneous differential equa-
tions. The general solution of a non-homogeneous second order equation. Complementary
functions and particular integrals.
2. The homogeneous equation with constant coefficients. auxiliary equation. The general
solution corresponding to real and distinct roots, real and equal roots and complex roots
of the auxiliary equation.
3. Non-homogeneous equations: The method of undetermined coefficients. The method of
variation of parameters.
Unit III: Linear System of ODEs (15 Lectures)
Existence and uniqueness theorems to be stated clearly when needed in the sequel. Study of
homogeneous linear system of ODEs in two variables: Let a1(t), a2(t), b1(t), b2(t) be continuous
real valued functions defined on [a, b]. Fix t0 ∈ [a, b]. Then there exists a unique solution
x = x(t), y = y(t) valid throughout [a, b] of the following system:
14
dx
dt
= a1(t)x+ b1(t)y,
dy
dt
= a2(t)x+ b2(t)y
satisfying the initial conditions x(t0) = x0&y(t0) = y0.
The Wronskian W (t) of two solutions of a homogeneous linear system of ODEs in two variables,
result: W (t) is identically zero or nowhere zero on [a, b]. Two linearly independent solutions
and the general solution of a homogeneous linear system of ODEs in two variables.
Explicit solutions of Homogeneous linear systems with constant coefficients in two variables,
examples.
Recommended Text Books for Unit I and II:
1. G. F. Simmons, Differential equations with applications and historical notes, McGraw Hill.
2. E. A. Coddington, An introduction to ordinary differential equations, Dover Books.
Recommended Text Book for Unit III:
G. F. Simmons, Differential equations with applications and historical notes, McGraw Hill.
USMT P04/UAMT P04 Practicals.
Suggested Practicals for USMT401/UAMT401:
1. Calculation of upper sum, lower sum and Riemann integral.
2. Problems on properties of Riemann integral.
3. Problems on fundamental theorem of calculus, mean value theorems, integration by parts,
Leibnitz rule.
4. Convergence of improper integrals, applications of comparison tests, Abel’s and Dirichlet’s
tests, and functions.
5. Beta Gamma Functions
6. Problems on area, volume, length.
7. Miscellaneous Theoretical Questions based on full paper.
Suggested Practicals for USMT402/UAMT 402:
1. Examples and properties of groups.
2. Group of symmetry of equilateral triangle, rectangle, square.
3. Subgroups.
4. Cyclic groups, cyclic subgroups, finding generators of every subgroup of a cyclic group.
5. Left and right cosets of a subgroup, Lagrange’s Theorem.
15
6. Group homomorphisms, isomorphisms.
7. Miscellaneous Theoretical questions based on full paper.
Suggested Practicals for USMT403:
1. Solving exact and non exact equations.
2. Linear and reducible to linear equations, applications to orthogonal trajectories, population
growth, and finding the current at a given time.
3. Finding general solution of homogeneous and non-homogeneous equations, use of known
solutions to find the general solution of homogeneous equations.
4. Solving equations using method of undetermined coefficients and method of variation of
parameters.
5. Solving second order linear ODEs
6. Solving a system of first order linear ODES.
7. Miscellaneous Theoretical questions from all units.
Scheme of Examination
I. Semester End Theory Examinations: There will be a Semester-end external Theory
examination of 100 marks for each of the courses USMT301/UAMT301, USMT302/UAMT302,
USMT303 of Semester III and USMT401/UAMT401, USMT402/UAMT402, USMT403 of
semester IV to be conducted by the University.
1. Duration: The examinations shall be of 3 Hours duration.
2. Theory Question Paper Pattern:
a) There shall be FIVE questions. The first question Q1 shall be of objective type
for 20 marks based on the entire syllabus. The next three questions Q2, Q2, Q3
shall be of 20 marks, each based on the units I, II, III respectively. The fifth
question Q5 shall be of 20 marks based on the entire syllabus.
b) All the questions shall be compulsory. The questions Q2, Q3, Q4, Q5 shall have
internal choices within the questions. Including the choices, the marks for each
question shall be 30-32.
c) The questions Q2, Q3, Q4, Q5 may be subdivided into sub-questions as a, b, c,
d & e, etc and the allocation of marks depends on the weightage of the topic.
d) The question Q1 may be subdivided into 10 sub-questions of 2 marks each.
II. Semester End Examinations Practicals:
At the end of the Semesters III and IV, Practical examinations of three hours duration
and 150 marks shall be conducted for the courses USMTP03, USMTP04.
At the end of the Semesters III and IV, Practical examinations of three hours duration
and 150 marks shall be conducted for the courses UAMTP03, UAMTP04.
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In semester III, the Practical examinations for USMT301/UAMT301 and USMT302/UAMT302
are held together by the college. The Practical examination for USMT303 is held sepa-
rately by the college.
In semester IV, the Practical examinations for USMT401/UAMT401 and USMT402/UAMT402
are held together by the college. The Practical examination for USMT403 is held sepa-
rately by the college.
Paper pattern: The question paper shall have three parts A, B, C.
Each part shall have two Sections.
Section I Objective in nature: Attempt any Eight out of Twelve multiple choice ques-
tions. (8× 3 = 24 Marks)
Section II Problems: Attempt any Two out of Three. (8× 2 = 16 Marks)
Practical Part A Part B Part C Marks duration
Course out of
USMTP03 Questions Questions Questions 120 3 hours
from USMT301 from USMT302 from USMT303
UAMTP03 Questions Questions — 80 2 hours
from UAMT301 from UAMT302
USMTP04 Questions Questions Questions 120 3 hours
from USMT401 from USMT402 from USMT403
UAMTP03 Questions Questions — 80 2 hours
from UAMT401 from UAMT402
Marks for Journals and Viva:
For each course USMT301/UAMT301, USMT302/UAMT302, USMT303, USMT401/UAMT401,
USMT402/UAMT402 and USMT403:
1. Journals: 5 marks.
2. Viva: 5 marks.
Each Practical of every course of Semester III and IV shall contain 10 (ten) problems out of
which minimum 05 (five) have to be written in the journal. A student must have a certified
journal before appearing for the practical examination.