Mathematics (Civil Engineering -1) | 1. Linear Differential Equations by Abraham | Learn Smarter
K12 Students

Academics

AI-Powered learning for Grades 8–12, aligned with major Indian and international curricula.

Professionals

Professional Courses

Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.

Games

Interactive Games

Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.

Typing
Memory
Math
English Adventures
Knowledge
1. Linear Differential Equations

The chapter introduces linear differential equations, emphasizing their importance in modeling various engineering problems. It covers first-order and second-order linear differential equations, classification based on order and degree, and standard methods to solve them, including the integrating factor and methods for finding particular solutions. The applications of these equations in civil engineering fields such as structural analysis, fluid mechanics, and geotechnical engineering are also highlighted.

Enroll to start learning

You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.

Sections

  • 1

    Linear Differential Equations

    Linear differential equations are essential tools in civil engineering for modeling various phenomena such as fluid flow and structural behavior.

    Learning Practice

  • 1.1

    Introduction

    This section introduces linear differential equations and their significance in civil engineering applications.

    Learning Practice

  • 1.2

    Basic Concepts And Terminology

    This section introduces fundamental concepts and terminology related to linear differential equations, including definitions, order, degree, and what constitutes a linear differential equation.

    Learning Practice

  • 1.2.1

    Differential Equation

    Differential equations are mathematical equations that relate an unknown function to its derivatives and are fundamental tools in engineering analysis.

    Learning Practice

  • 1.2.2

    Order And Degree

    This section explains the fundamental concepts of order and degree of differential equations, which are crucial for classifying and solving such equations.

    Learning Practice

  • 1.2.3

    Linear Differential Equation

    Linear differential equations involve dependent variables and their derivatives to the first power, forming the backbone of many engineering applications.

    Learning Practice

  • 1.3

    First-Order Linear Differential Equations

    This section covers the general form and solution method for first-order linear differential equations, highlighting the integrating factor approach.

    Learning Practice

  • 1.3.1

    General Form

    This section introduces the general form of first-order linear differential equations, specifically the equation structure and how it is applied in engineering contexts.

    Learning Practice

  • 1.3.2

    Solution Method: Integrating Factor (If)

    The Integrating Factor method is a crucial approach for solving first-order linear differential equations.

    Learning Practice

  • 1.3.3

    Example

    This section provides an example of solving a first-order linear differential equation using the integrating factor method.

    Learning Practice

  • 1.4

    Second-Order Linear Differential Equations

    This section introduces second-order linear differential equations, their general forms, and classifications, alongside methods to solve them.

    Learning Practice

  • 1.4.1

    General Form

    The General Form of second-order linear differential equations outlines the structure of the equations and differentiates between homogeneous and non-homogeneous types.

    Learning Practice

  • 1.5

    Homogeneous Equations With Constant Coefficients

    This section covers the general forms and solutions of homogeneous linear differential equations with constant coefficients.

    Learning Practice

  • 1.5.1

    General Form

    This section introduces the general form of homogeneous linear differential equations with constant coefficients and their auxiliary equations.

    Learning Practice

  • 1.5.2

    Auxiliary Equation (Ae)

    The Auxiliary Equation (AE) is utilized to solve homogeneous linear differential equations with constant coefficients by finding the roots, which determine the general solution forms.

    Learning Practice

  • 1.5.3

    Example

    This section provides a practical example of solving a second-order homogeneous linear differential equation with constant coefficients.

    Learning Practice

  • 1.6

    Non-Homogeneous Linear Equations

    This section introduces non-homogeneous linear equations, explaining their general form and solution methods, including complementary and particular solutions.

    Learning Practice

  • 1.6.1

    General Form

    The general form of non-homogeneous linear differential equations is crucial for understanding engineering applications.

    Learning Practice

  • 1.6.2

    Complete Solution

    The Complete Solution in linear differential equations involves finding both the complementary function and the particular solution.

    Learning Practice

  • 1.7

    Methods Of Finding Particular Solution

    This section discusses the methods for finding particular solutions to non-homogeneous linear differential equations, specifically the Method of Undetermined Coefficients and the Method of Variation of Parameters.

    Learning Practice

  • 1.7.1

    Method Of Undetermined Coefficients

    The Method of Undetermined Coefficients is a technique used to find particular solutions to non-homogeneous linear differential equations.

    Learning Practice

  • 1.7.2

    Method Of Variation Of Parameters

    The method of variation of parameters is a technique used to find particular solutions to non-homogeneous linear differential equations when the standard method of undetermined coefficients is not applicable.

    Learning Practice

  • 1.8

    Fourier Transform (Overview In Engineering Context)

    The Fourier Transform provides a method to convert differential equations into algebraic equations, aiding in solving engineering problems.

    Learning Practice

  • 1.9

    Applications In Civil Engineering

    This section outlines various applications of linear differential equations in civil engineering, including structural analysis and fluid mechanics.

    Learning Practice

  • 1.10

    Summary

    Linear differential equations are fundamental in engineering for analyzing and solving various physical problems.

    Learning Practice

References

Chapter_1_Linear.pdf

Class Notes

Memorization

What we have learnt

  • Linear differential equatio...
  • The methods for solving fir...
  • Applications in civil engin...

Final Test

Revision Tests