Unit 1 Differential Equations

8 lecture hours

• Basic Concepts: Modelling
• Differential Equations
• Ordinary and Partial differentiation
• Order of the equation
• Existence and Uniqueness of Solution
• Initial Value problem
• Boundary Value Problem
• Linear and Non-Linear Equation. 1st Order ODE: Geometric Meaning of y'= ƒ(x
• Direction Fields
• Euler’s Method
• Separable ODEs
• Exact ODEs (Integrating Factors Method
• Linear ODEs(Homogeneous and Non-Homogeneous
• Reduction to Linear problems)
• Orthogonal Trajectories. 2nd Order ODE: Linear Dependence and Linear Independence
• Homogeneous Linear ODEs of Second Order (Principal of Superposition
• Homogeneous Linear ODEs with Constant Coefficients (Euler's formula and review of the circular and hyperbolic function
• Exponential Solutions
• Repeated Roots and Stability)
• Differential Operator
• Modelling of Free Oscillations of Spring-Mass System
• Homogeneous Linear ODEs with Non-constant Coefficient (Cauchy-Euler Equation
• Existence and Uniqueness of Solutions)

Unit 2 Laplace Transforms

5 lecture hours

• Laplace Transform
• First Shifting Theorem (s-Shifting)
• Transforms of Derivatives and Integrals, ODE
• Unit Step Function (Heaviside Function)
• Second Shifting Theorem (t-Shifting)
• Short Impulses
• Dirac’s Delta Function
• Partial Fractions
• Integral Equations
• Differentiation and Integration of Transforms
• ODEs with Variable Coefficients
• Systems of ODEs

Unit 3 Linear Algebra

• Matrices and Vectors: Vectors in 2-Space and 3-Space
• Addition and Scalar Multiplication
• Matrix Multiplication
• Linear Systems of Equations and Gauss Elimination
• Ill- Conditioning
• Linear Independence
• Rank of a Matrix
• Solutions of Linear Systems: Existence 07 Subject Code: 3710812 w.e.f
• AY 2018-19 # Exam paper should be based on applications of above topics in Mechanical engineering rather than merely derivation of standard theory or its mathematical solution