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# 1st Order Differential Equations

149.00

This module has a detailed introduction to engineering relevance of differential equations. The module also covers linear and non-linear 1st order differential equations. Significant number of model questions paper problems have been solved to give a comprehensive understanding of the examiner’s perspective.

Salient features of this module are:

• Pre-requisites to Elements of Differential & Integral Calculus
• Graphical visualization
• Concept testing
• Unravelling examiners mind
• Memory structure (for derivations)
• Model question paper

Course Contents:

Pre-requisites to Maxima & Minima

• Sign of Trigonometric Identities
• Sign of Trigonometric Identities – Introduction
• Sign of Trigonometric Identities – Quad I
• Sign of Trigonometric Identities – Quad II
• Sign of Trigonometric Identities – Quad III
• Sign of Trigonometric Identities – Quad IV
• Sign of Trigonometric Identities – Summary
• Sign of Trigonometric Identities – 1
• Sign of Trigonometric Identities – 2
• General solution for Trigonometric  ratios
• General solution for Trigonometric ratios (Example 1)
• General solution for Trigonometric ratios (Example 2)
• General solution for Trigonometric ratios (Example 3)
• General solution for Trigonometric ratios (Example 4)
• Concept of derivate – Slope
• Concept of Derivative – Slope – Straight line
• Concept of Derivative – Slope – Curve
• Differentiation of Basic functions
• Introduction – Chain and Quotient Rules
• Derivative of tan x
• Derivative of sec x
• Derivative of cosec x
• Derivative of cot x
• Derivative of [y = e^x * x^n]
• Derivative of [y = tan^(-1) x
• Derivative of [y = x/(ax+b)] – Quotient rule
• Derivative of [y = x/(ax+b)] – Chain rule
• Derivative of [x^2 + y^2 = k]
• Derivative of [y = 1/sqrt(ax^2 +b)]
• Derivative of [y = a^x (log (1+x^2))]
• Derivative of [y = a^sinx (1+sin^2 (x)) sec x]
• Graphical understanding of trigonometric derivatives
• Application of Derivative
• Finding nature of the curve
• Interpreting curvature of a curve
• Linearization of a curve
• L-Hospital Rule
• Use of second derivative (curvature)
• Maxima & Minima – 1
• Maxima & Minima – 2
• Angle between Curves & Radius of curvature
• Essential trigonometric identities 1
• Essential trigonometric identities 2
• Essential trigonometric identities 3
• Essential trigonometric identities 4
• Essential trigonometric identities 5

PRE-REQUISITES – ELEMENTS OF INTEGRAL CALCULUS

• Pre-requisites – Elements of Integral Calculus
• Understanding Integration
• Concept of Integration – Elementary Example
• Introducing the Learning matrix
• Why is Integration an anti-differentiation process?
• Useful Rules and Symbolisms of Integration – (Rule 1)
• Useful Rules and Symbolisms of Integration – (Rule 2)
• Introducing Standard Integrals
• Deriving Integrals of Basic function
• Deriving Specific Integrals – 1
• Deriving Specific Integrals – 2
• Deriving Specific Integrals – 3
• Deriving Specific Integrals – 4
• Deriving Specific Integrals – 5
• Useful Rules of Definite Integrals – (Rule 1)
• Useful Rules of Definite Integrals – (Rule 2)
• Useful Rules of Definite Integrals – (Rule 3)
• Useful Rules of Definite Integrals – (Rule 4)

1ST ORDER DIFFERENTIAL EQUATION

• Engineering Relevance
• Understanding Formation of DE’s
• Predicting the System Behavior
• Testing System Behavior
• Complexities in Solving DE’s
• Need for PDE’s for Solving DE’s
• Understanding Non-linearity in DE’s
• A Brief Histor of DE’s
• Differential Equation(1st order)
• Understanding the Representations of DE’s
• Mathematical Observation of DE’s
• Deriving the Integrating Factor (IF)
• Linear and Bernoulli’s DE’s – Problem
• Problem 1
• Problem 2
• Problem 3
• Problem 4
• Problem 5
• Problem 6
• Problem 7
• Problem 8
• Unravelling the Examiner’s Mind
• Non – Linear Differential Equation
• Engineering Relevance
• Introduction
• Non – Linear Differential Equation – Problems
• Problem 1 – ‘p’ type
• Problem 2 – ‘p’ type
• Problem 3 – ‘Clairaut’ type
• Problem 4 – ‘Clairaut’ type
• Problem 5 – ‘Reducible Clairaut’ type
• Problem 6 – ‘Reducible Clairaut’ type
• Unravelling Examiner’s Mind

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