Syllabus (MATHEMATICS)

Course Type: MAJ-3

Semester: 3

Course Code: BMTMMAJ03T

Course Title: Ordinary Differential Equations & Linear Algebra-I

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: TO BE DONE

Learning Outcome: The whole course will have the following outcomes: Upon successful completion, student will be able to CO1: solve first order differential equations utilizing the standard techniques for exact, linear, homogeneous, or Bernoulli cases. CO2: solve first ord

Syllabus:

Unit -1: Ordinary Differential Equations [Credit: 3, 45L]

• Prerequisite [Genesis of differential equation: Order, degree and solution of an ordinary differential equation, Formation of ODE, Meaning of the solution of ordinary differential equation, Concept of linear and non-linear differential equations].
• Picard’s existence and uniqueness theorem (statement only) for dydx=f(x, y) with y = y0 at x = x0 and its applications.
• Solution of Differential Equations of first order and first degree: Homogeneous equations and equations reducible to homogenous form. Exact differential equations, condition of exactness, Integrating Factor, Rules of finding integrating factor (statement of relevant results only), equations reducible to exact forms, Linear Differential Equations, equations reducible to linear forms, Bernoulli’s equations.
• Solution of Differential Equations of first order but not of first degree: Equations solvable for p, equations solvable for x, equation solvable for y, singular solutions, Clairaut’s form, equations reducible to Clairaut’s Forms - General and Singular solutions.

Applications of first order differential equations: Geometric applications, Orthogonal Trajectories.

• Solution of Differential Equations of higher order: Linear differential equations of second and higher order, Linearly dependent and independent solutions, Wronskian, General solution of second order linear differential equation, General and particular solution of linear differential equation of second order with constant coefficients. Particular integrals for polynomial, sine, cosine, exponential function and for function as combination of them or involving them, Method of variation of parameters for particular integral (P.I.) of linear differential equation of second order
• Linear Differential Equations with variable co-efficients, Euler- Cauchy equations, Exact differential equations, Reduction of order of linear differential equation, Reduction to normal form. Solutions of some special types of differential equations.
• Simultaneous linear ordinary differential equation in two dependent variables, Solution of simultaneous equations of the form dx/P = dy/Q = dz/R.
• Pfaffian Differential Equation Pdx +Qdy+Rdz = 0, Necessary and sufficient condition for existence of integrals of the above (proof not required), Total differential equation.

Unit -2: Linear Algebra-I [Credit: 3, 45L]

• Vector space, subspaces, Linear Sum, linear span, linearly dependent and independent vectors, basis, dimension, finite dimensional vector spaces, Replacement Theorem, Extension theorem, Deletion theorem, change of coordinates, Row rank and column rank of a matrix. Rank of a matrix. Row reduced echelon matrix, Normal form of the matrix.
• Systems of linear equations and the matrix equation Ax=b. Existence of solutions of homogeneous and non-homogeneous system of equations and determination of their solutions.
• Characteristic Equation of a matrix, Cayley-Hamilton theorem (statement only) and its applications. Eigen values, Eigen Vectors of a matrix, Similar matrices, Diagonalization of matrices of order 2 and 3.

Reading References:

1. S.L. Ross, Differential Equations, 3rd Ed., John Wiley and Sons, India, 2004.
2. Murray, D., Introductory Course in Differential Equations, Longmans Green and Co.
3. Boyce and Diprima, Elementary Differential Equations and Boundary Value Problems, Wiley.
4. G.F. Simmons, Differential Equations, Tata McGraw Hill
5. Ghosh and Chakraborty, Differential Equations, U N Dhur
6. Maity and Ghosh, An Introduction to Differential Equations, New Central Book Agency
7. G. C. Gorain, Introductory course on Differential Equations
8. S. K. Mapa, Higher Algebra, Vol. – II, Academic Publishers
9. Hoffman, K. and Kunze, R., Linear Algebra, Pearson
10. Finkbeincr, Matrices and Linear Algebra