Engineering Mathematics-II (BTAM201-23)

Course Objectives

The objective of Engineering Mathematics-II is to enable students to understand concepts like:

• Elimination methods for solving linear systems of equations
• Inverse of matrices
• Vector spaces
• Ordinary and partial differential equations

The goal is to equip students with the ability to apply these techniques to solve engineering problems.

Course Outcomes

• Verify theoretical concepts through practical application.
• Visualize and conceptualize engineering problems.
• Mathematically model engineering problems using matrices, ODEs, and PDEs.
• Draw conclusions from data and develop skills applicable to industrial problems.
• Find solutions to engineering problems from an application-oriented perspective.

Detailed Syllabus

PART-A

Unit-I: System of Linear Equations

• Rank of a matrix
• Echelon form of matrix
• Homogeneous and Non-homogeneous systems of linear equations
• Consistency and inconsistency of systems of equations
• Gauss elimination method
• Inverse of a matrix
• Gauss-Jordan method

Unit-II: Vector Spaces

• Vector spaces and Subspaces
• Linear independence and dependence of vectors
• Dimension and basis
• Linear transformation, rank and nullity theorem (without proof)
• Matrix associated with Linear Transformation
• Eigenvalues and eigenvectors
• Cayley-Hamilton theorem
• Algebraic and geometric multiplicity
• Similar and diagonalizable matrices

PART-B

Unit-III: Ordinary Differential Equations (ODEs)

• Formation and solution of differential equations
• Initial and boundary value problems
• Solution methods:
  • Separable equations
  • Equations reducible to separable form
  • Exact differential equations
  • Integrating factors
  • Linear first-order equations
  • Bernoulli and Riccati equations
  • Clairaut's equation
  • Higher-order linear equations (constant & variable coefficients)
  • Variation of parameters
  • Undetermined coefficients
  • Finding particular integrals
• Applications:
  • Electric RLC circuits
  • Beam deflection
  • Simple harmonic motion
  • Population decay models
  • Orthogonal trajectories

Unit-IV: Partial Differential Equations (PDEs)

• Formation of first and second-order PDEs
• Solution of first-order PDEs:
  • Lagrange's equation
  • Surfaces orthogonal to a given family
  • Nonlinear first-order equations
  • Charpit's method
• Higher-order linear PDEs with constant coefficients