Engineering Mathematics Exam > Engineering Mathematics Notes > Differential Equation and Mathematical Modeling-II > Lecture 7 - First Order Partial Differential Equations
Lecture 7 - First Order Partial Differential Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
Page 2
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter : First Order Partial Differential Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Classification of partial differential equations of order one
? 3.1: Linear partial differential equations
? 3.2: Semi linear partial differential equations
? 3.3: Quasi linear partial differential equations
? 3.4: Non linear partial differential equations
? 4: Construction of partial differential equations of first order
? 4.1: Elimination of arbitrary constants
o Exercises
? 4.2: By elimination of arbitrary functions
o Exercises
? 5: Classification of Integrals or Solutions
? 6: Geometrical interpretation of partial differential equation of
first order
? 7: Method of Characteristics
? 8: The Cauchy problem for partial differential equation of first
order
o Exercises
? 9: Canonical Forms of Linear Equations of First Order
o Exercise
? 10: Method of Separation of Variables
o Exercises
? Summary
? Glossary
? References/ Further Reading
Page 3
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter : First Order Partial Differential Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Classification of partial differential equations of order one
? 3.1: Linear partial differential equations
? 3.2: Semi linear partial differential equations
? 3.3: Quasi linear partial differential equations
? 3.4: Non linear partial differential equations
? 4: Construction of partial differential equations of first order
? 4.1: Elimination of arbitrary constants
o Exercises
? 4.2: By elimination of arbitrary functions
o Exercises
? 5: Classification of Integrals or Solutions
? 6: Geometrical interpretation of partial differential equation of
first order
? 7: Method of Characteristics
? 8: The Cauchy problem for partial differential equation of first
order
o Exercises
? 9: Canonical Forms of Linear Equations of First Order
o Exercise
? 10: Method of Separation of Variables
o Exercises
? Summary
? Glossary
? References/ Further Reading
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning Outcomes:
After reading this chapter, learner will be able to understand
? How to construct a first order PDE
? How to solve a first order PDE using the method of characteristics
? How to transform a PDE of first order in canonical form.
? How to solve PDE of first order using the method separation of
variables.
2. Introduction:
Page 4
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter : First Order Partial Differential Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Classification of partial differential equations of order one
? 3.1: Linear partial differential equations
? 3.2: Semi linear partial differential equations
? 3.3: Quasi linear partial differential equations
? 3.4: Non linear partial differential equations
? 4: Construction of partial differential equations of first order
? 4.1: Elimination of arbitrary constants
o Exercises
? 4.2: By elimination of arbitrary functions
o Exercises
? 5: Classification of Integrals or Solutions
? 6: Geometrical interpretation of partial differential equation of
first order
? 7: Method of Characteristics
? 8: The Cauchy problem for partial differential equation of first
order
o Exercises
? 9: Canonical Forms of Linear Equations of First Order
o Exercise
? 10: Method of Separation of Variables
o Exercises
? Summary
? Glossary
? References/ Further Reading
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning Outcomes:
After reading this chapter, learner will be able to understand
? How to construct a first order PDE
? How to solve a first order PDE using the method of characteristics
? How to transform a PDE of first order in canonical form.
? How to solve PDE of first order using the method separation of
variables.
2. Introduction:
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 4
An equation is called a partial differential equation if it contains one or more
partial derivatives. In the 18th century, Lagrange, Laplace D'Alembert and
Euler started the study of partial differential equations (PDE's) as the
principal mode of analytical study of models in the physical science. In the
middle of the 19
th
century, PDE's also became a necessary tool in other
branches of mathematics.
3. Classification of partial differential equations of order
one:
The most general form of the partial differential equations of order one in
two independent variables and is given by
(
)
Or
( ) (1)
Where is a function of two independent variables , and dependent
variable and its partial derivatives
.
Similarly, if there are three independent variables then the partial
differential equations of order one can be written as
(
) (2)
3.1. Linear partial differential equations:
A partial differential equation ( ) is said to be linear if the
degree of the dependent variable and all its partial derivatives i.e. is
one.
i.e. if it is of the form
( )
( )
( ) ( )
Where ( ) ( ) ( ) and ( ) are functions of only.
3.1.1. Examples of linear partial differential equations:
(a)
( )
Page 5
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter : First Order Partial Differential Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Classification of partial differential equations of order one
? 3.1: Linear partial differential equations
? 3.2: Semi linear partial differential equations
? 3.3: Quasi linear partial differential equations
? 3.4: Non linear partial differential equations
? 4: Construction of partial differential equations of first order
? 4.1: Elimination of arbitrary constants
o Exercises
? 4.2: By elimination of arbitrary functions
o Exercises
? 5: Classification of Integrals or Solutions
? 6: Geometrical interpretation of partial differential equation of
first order
? 7: Method of Characteristics
? 8: The Cauchy problem for partial differential equation of first
order
o Exercises
? 9: Canonical Forms of Linear Equations of First Order
o Exercise
? 10: Method of Separation of Variables
o Exercises
? Summary
? Glossary
? References/ Further Reading
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning Outcomes:
After reading this chapter, learner will be able to understand
? How to construct a first order PDE
? How to solve a first order PDE using the method of characteristics
? How to transform a PDE of first order in canonical form.
? How to solve PDE of first order using the method separation of
variables.
2. Introduction:
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 4
An equation is called a partial differential equation if it contains one or more
partial derivatives. In the 18th century, Lagrange, Laplace D'Alembert and
Euler started the study of partial differential equations (PDE's) as the
principal mode of analytical study of models in the physical science. In the
middle of the 19
th
century, PDE's also became a necessary tool in other
branches of mathematics.
3. Classification of partial differential equations of order
one:
The most general form of the partial differential equations of order one in
two independent variables and is given by
(
)
Or
( ) (1)
Where is a function of two independent variables , and dependent
variable and its partial derivatives
.
Similarly, if there are three independent variables then the partial
differential equations of order one can be written as
(
) (2)
3.1. Linear partial differential equations:
A partial differential equation ( ) is said to be linear if the
degree of the dependent variable and all its partial derivatives i.e. is
one.
i.e. if it is of the form
( )
( )
( ) ( )
Where ( ) ( ) ( ) and ( ) are functions of only.
3.1.1. Examples of linear partial differential equations:
(a)
( )
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 5
(b)
(c) (
)
(
)
( ) ( )
3.2. Semi linear partial differential equations:
A partial differential equation of order one ( ) is said to be semi
linear if it is of the form
( )
( )
( )
Where and are the functions of only.
3.2.1. Examples of semi linear partial differential equations:
(a)
(b) ( )
( )
( )
(c)
( )
3.3. Quasi linear partial differential equations:
A partial differential equation ( ) is said to be quasi linear if the
degree of the all partial derivatives i.e. is one. i.e. if it is of the form
( )
( )
( )
Where and are the functions of .
3.3.1. Examples of quasi linear partial differential equations:
(a) (
)
(b) (
)
(c) (
)
(
)
3.4. Non linear partial differential equations:
A partial differential equation ( ) is said to be non linear if the
degree of the partial derivatives i.e. is not one.
3.4.1. Examples of non-linear partial differential equations:
(a)
(b)
(
)
(c)
Read More
FAQs on Lecture 7 - First Order Partial Differential Equations - Differential Equation and Mathematical Modeling-II - Engineering Mathematics
| 1. What are first-order partial differential equations? |  |
Ans. First-order partial differential equations are mathematical equations that involve partial derivatives of an unknown function with respect to one or more independent variables. They are called "first-order" because they involve only first derivatives and not higher-order derivatives.
| 2. What is the significance of first-order partial differential equations in engineering mathematics? | |
Ans. First-order partial differential equations have significant applications in engineering mathematics. They are used to model various physical phenomena, such as heat transfer, fluid flow, and electromagnetic fields. Solving these equations helps engineers understand and predict the behavior of these systems, enabling them to design better solutions.
| 3. How do we solve first-order partial differential equations? | |
Ans. The solution to a first-order partial differential equation involves finding a function that satisfies the equation. Different techniques can be used depending on the specific equation. Common methods include the method of characteristics, separation of variables, and the use of integral transforms such as the Laplace transform or Fourier transform.
| 4. Can you provide an example of a first-order partial differential equation in engineering mathematics? | |
Ans. Certainly! One example of a first-order partial differential equation is the advection equation, which describes the transport of a quantity by a moving medium. It can be written as ∂u/∂t + c∂u/∂x = 0, where u is the unknown quantity, t is time, x is the spatial variable, and c is the velocity of the medium.
| 5. Are there any real-life applications of first-order partial differential equations in engineering? | |
Ans. Yes, first-order partial differential equations find numerous applications in engineering. For example, they are used to model the flow of fluids in pipes, analyze heat conduction in solid materials, simulate electrical circuits, and study the propagation of sound waves. These equations allow engineers to optimize designs, predict behavior, and solve practical problems in various fields of engineering.
Related Exams
About this Document
|
1.2K Views |
|
4.89/5 Rating |
|
Dec 23, 2024 Last updated |
Document Description: Lecture 7 - First Order Partial Differential Equations for Engineering Mathematics 2024 is part of Differential Equation and Mathematical Modeling-II preparation.
The notes and questions for Lecture 7 - First Order Partial Differential Equations have been prepared according to the Engineering Mathematics exam syllabus. Information about Lecture 7 - First Order Partial Differential Equations covers topics
like and Lecture 7 - First Order Partial Differential Equations Example, for Engineering Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Lecture 7 - First Order Partial Differential Equations.
Introduction of Lecture 7 - First Order Partial Differential Equations in English is available as part of our Differential Equation and Mathematical Modeling-II
for Engineering Mathematics & Lecture 7 - First Order Partial Differential Equations in Hindi for Differential Equation and Mathematical Modeling-II course.
Download more important topics related with notes, lectures and mock test series for Engineering Mathematics
Exam by signing up for free. Engineering Mathematics: Lecture 7 - First Order Partial Differential Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics
Description
Full syllabus notes, lecture & questions for Lecture 7 - First Order Partial Differential Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics - Engineering Mathematics | Plus excerises question with solution to help you revise complete syllabus for Differential Equation and Mathematical Modeling-II | Best notes, free PDF download
Information about Lecture 7 - First Order Partial Differential Equations
In this doc you can find the meaning of Lecture 7 - First Order Partial Differential Equations defined & explained in the simplest way possible. Besides explaining types of
Lecture 7 - First Order Partial Differential Equations theory, EduRev gives you an ample number of questions to practice Lecture 7 - First Order Partial Differential Equations tests, examples and also practice Engineering Mathematics
tests
|
Explore Courses for Engineering Mathematics exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Lecture 7 - First Order Partial Differential Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics
,mock tests for examination
,Previous Year Questions with Solutions
,shortcuts and tricks
,study material
,video lectures
,Objective type Questions
,Important questions
,Exam
,MCQs
,ppt
,past year papers
,Viva Questions
,Lecture 7 - First Order Partial Differential Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics
,Free
,Sample Paper
,Semester Notes
,Lecture 7 - First Order Partial Differential Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics
,practice quizzes
,Extra Questions
,Summary
;
Additional Information about Lecture 7 - First Order Partial Differential Equations for Engineering Mathematics Preparation
Lecture 7 - First Order Partial Differential Equations Free PDF Download
The Lecture 7 - First Order Partial Differential Equations is an invaluable resource that delves deep into the core of the Engineering Mathematics exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Lecture 7 - First Order Partial Differential Equations now and kickstart your journey towards success in the Engineering Mathematics exam.
Importance of Lecture 7 - First Order Partial Differential Equations
The importance of Lecture 7 - First Order Partial Differential Equations cannot be overstated, especially for Engineering Mathematics aspirants.
This document holds the key to success in the Engineering Mathematics exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Lecture 7 - First Order Partial Differential Equations Notes
Lecture 7 - First Order Partial Differential Equations Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Lecture 7 - First Order Partial Differential Equations.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Lecture 7 - First Order Partial Differential Equations Notes on EduRev are your ultimate resource for success.
Lecture 7 - First Order Partial Differential Equations Engineering Mathematics Questions
The "Lecture 7 - First Order Partial Differential Equations Engineering Mathematics Questions" guide is a valuable resource for all aspiring students preparing for the
Engineering Mathematics exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Lecture 7 - First Order Partial Differential Equations on the App
Students of Engineering Mathematics can study Lecture 7 - First Order Partial Differential Equations alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Lecture 7 - First Order Partial Differential Equations,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Lecture 7 - First Order Partial Differential Equations is prepared as per the latest Engineering Mathematics syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup