UPSC Exam > UPSC Notes > Mathematics Optional Notes for UPSC > Formulation of Differential Equations
Formulation of Differential Equations | Mathematics Optional Notes for UPSC PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Page 2
Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Alternatively: We can also use equation of circle ?? 2
+(?? -1)
2
+?? (?? -1)=0 and
proceed.
1.2 Find the curve for which the part of the tangent cut-off by the axes is bisected
at the point of tangency.
(2014 : 10 Marks)
Solution:
Let equation of tangent line at point ' ?? ' al
?? -?? ?? -?? =
????
????
(??)
Now, its point of intersection with co-ordinate axes are
?? (0,?? -
?? ????
????
);?? (?? -?? ????
????
,0)
Given : ' ?? ' is mid point of ???? .
So,
Page 3
Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Alternatively: We can also use equation of circle ?? 2
+(?? -1)
2
+?? (?? -1)=0 and
proceed.
1.2 Find the curve for which the part of the tangent cut-off by the axes is bisected
at the point of tangency.
(2014 : 10 Marks)
Solution:
Let equation of tangent line at point ' ?? ' al
?? -?? ?? -?? =
????
????
(??)
Now, its point of intersection with co-ordinate axes are
?? (0,?? -
?? ????
????
);?? (?? -?? ????
????
,0)
Given : ' ?? ' is mid point of ???? .
So,
?? -?? ????
????
2
=?? and
?? -
?? ????
2
=??
? ?? =-?? ????
????
and ?? =-
?? ????
????
?
????
????
=-
?? ??
? ?????? +?????? =0
Integrating, we get
?? 2
+?? 2
=?? which the required curve.
1.3 Find the differential equation (DE) representing all the circles in the ???? -plane.
(2017 : 10 Marks)
Solution:
Method 1: General equation of circle
(?? -?? )
2
+(?? -?? )
2
=?? 2
Differentiating w.r.t. ?? ,
2(?? -?? )+2(?? -?? )
????
????
=0
??.?? ., (?? -?? )+(?? -?? )?? 1
=0 (??)
Differentiating again w.r.t. ??
1+(?? -?? )?? 2
+?? 1
2
=0 (???? )
Differentiating again w.r.t. ??
(?? -?? )?? 3
+?? 1
?? 2
+2?? 1
?? 2
=0
i.e.,
(?? -?? )=
-3?? 1
?? 2
?? 3
Substituting it in (ii)
Page 4
Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Alternatively: We can also use equation of circle ?? 2
+(?? -1)
2
+?? (?? -1)=0 and
proceed.
1.2 Find the curve for which the part of the tangent cut-off by the axes is bisected
at the point of tangency.
(2014 : 10 Marks)
Solution:
Let equation of tangent line at point ' ?? ' al
?? -?? ?? -?? =
????
????
(??)
Now, its point of intersection with co-ordinate axes are
?? (0,?? -
?? ????
????
);?? (?? -?? ????
????
,0)
Given : ' ?? ' is mid point of ???? .
So,
?? -?? ????
????
2
=?? and
?? -
?? ????
2
=??
? ?? =-?? ????
????
and ?? =-
?? ????
????
?
????
????
=-
?? ??
? ?????? +?????? =0
Integrating, we get
?? 2
+?? 2
=?? which the required curve.
1.3 Find the differential equation (DE) representing all the circles in the ???? -plane.
(2017 : 10 Marks)
Solution:
Method 1: General equation of circle
(?? -?? )
2
+(?? -?? )
2
=?? 2
Differentiating w.r.t. ?? ,
2(?? -?? )+2(?? -?? )
????
????
=0
??.?? ., (?? -?? )+(?? -?? )?? 1
=0 (??)
Differentiating again w.r.t. ??
1+(?? -?? )?? 2
+?? 1
2
=0 (???? )
Differentiating again w.r.t. ??
(?? -?? )?? 3
+?? 1
?? 2
+2?? 1
?? 2
=0
i.e.,
(?? -?? )=
-3?? 1
?? 2
?? 3
Substituting it in (ii)
1+(
-3?? 1
?? 2
?? 3
)?? 2
+?? 1
2
=0
??.?? ., (1+?? 1
2
)?? 3
-3?? 1
?? 2
2
=0
??.?? ., (1+?? 1
2
)?? 3
=3?? 1
?? 2
2
Method II : Using curvature-formula (?? ) .
1.4 Find the orthogonal trajectories of the family of circles passing the points
(?? ,?? ) and (?? ,-?? ) .
(2020: 10 marks)
Solution:
Let equation of circle through (0,2) and (0,-2) be:
?? 2
+(?? 2
-4)+???? =0, ?? : Parameter (1)
Differentiating w.r.t. ?? , we get:
2?? +2?? (
????
????
)+?? =0 (2)
From (1) and (2),
?? 2
+?? 2
-4+[-2?? -2?? (
????
????
)]?? =0
?? 2
-?? 2
-4-2????
????
????
=0 (3)
Replace
????
????
by -
????
????
in (3), we get
?? 2
-?? 2
-4+2????
????
????
=0
(?? 2
-4)????
?? 2
+
2???????? -?? 2
????
?? 2
=0
? (1-
4
?? 2
)???? +? ?? (
?? 2
?? ) =0
?? +
4
?? +
?? 2
?? =0
?? 2
+?? 2
+4 =???? (required trajectory)
1.5 Find the orthogonal trajectories of the family of confocal conics
?? ?? ?? ?? +?? +
?? ?? ?? ?? +?? =?? ;?? >?? >??
Page 5
Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Alternatively: We can also use equation of circle ?? 2
+(?? -1)
2
+?? (?? -1)=0 and
proceed.
1.2 Find the curve for which the part of the tangent cut-off by the axes is bisected
at the point of tangency.
(2014 : 10 Marks)
Solution:
Let equation of tangent line at point ' ?? ' al
?? -?? ?? -?? =
????
????
(??)
Now, its point of intersection with co-ordinate axes are
?? (0,?? -
?? ????
????
);?? (?? -?? ????
????
,0)
Given : ' ?? ' is mid point of ???? .
So,
?? -?? ????
????
2
=?? and
?? -
?? ????
2
=??
? ?? =-?? ????
????
and ?? =-
?? ????
????
?
????
????
=-
?? ??
? ?????? +?????? =0
Integrating, we get
?? 2
+?? 2
=?? which the required curve.
1.3 Find the differential equation (DE) representing all the circles in the ???? -plane.
(2017 : 10 Marks)
Solution:
Method 1: General equation of circle
(?? -?? )
2
+(?? -?? )
2
=?? 2
Differentiating w.r.t. ?? ,
2(?? -?? )+2(?? -?? )
????
????
=0
??.?? ., (?? -?? )+(?? -?? )?? 1
=0 (??)
Differentiating again w.r.t. ??
1+(?? -?? )?? 2
+?? 1
2
=0 (???? )
Differentiating again w.r.t. ??
(?? -?? )?? 3
+?? 1
?? 2
+2?? 1
?? 2
=0
i.e.,
(?? -?? )=
-3?? 1
?? 2
?? 3
Substituting it in (ii)
1+(
-3?? 1
?? 2
?? 3
)?? 2
+?? 1
2
=0
??.?? ., (1+?? 1
2
)?? 3
-3?? 1
?? 2
2
=0
??.?? ., (1+?? 1
2
)?? 3
=3?? 1
?? 2
2
Method II : Using curvature-formula (?? ) .
1.4 Find the orthogonal trajectories of the family of circles passing the points
(?? ,?? ) and (?? ,-?? ) .
(2020: 10 marks)
Solution:
Let equation of circle through (0,2) and (0,-2) be:
?? 2
+(?? 2
-4)+???? =0, ?? : Parameter (1)
Differentiating w.r.t. ?? , we get:
2?? +2?? (
????
????
)+?? =0 (2)
From (1) and (2),
?? 2
+?? 2
-4+[-2?? -2?? (
????
????
)]?? =0
?? 2
-?? 2
-4-2????
????
????
=0 (3)
Replace
????
????
by -
????
????
in (3), we get
?? 2
-?? 2
-4+2????
????
????
=0
(?? 2
-4)????
?? 2
+
2???????? -?? 2
????
?? 2
=0
? (1-
4
?? 2
)???? +? ?? (
?? 2
?? ) =0
?? +
4
?? +
?? 2
?? =0
?? 2
+?? 2
+4 =???? (required trajectory)
1.5 Find the orthogonal trajectories of the family of confocal conics
?? ?? ?? ?? +?? +
?? ?? ?? ?? +?? =?? ;?? >?? >??
are constants and ?? is a parameter. Show that the given family of curves is self
orthogonal.
[2021 : 10 marks]
Solution:
(i)
Given:
?? 2
?? 2
+?? +
?? 2
?? 2
+?? =1
Differentiating (i)
2?? ?? 2
+?? +
2?? ?? 2
+?? ????
????
=0
or ?? (?? 2
+?? )+?? (?? 2
+?? )
????
????
=0
or,
?? (?? +?? ????
????
)=-(?? 2
?? +?? 2
?? ????
????
)
? ?? =-[
?? 2
?? +?? 2
?? (
????
????
)
?? +?? (
????
????
)
]
? ?? 2
+?? =?? 2
-
?? 2
?? +?? 2
?? (
????
????
)
?? +?? (
????
????
)
=
(?? 2
-?? 2
)?? ?? +?? (
?? ?? ?? ????
)
and
?? 2
+?? =?? 2
-
?? 2
?? +?? 2
?? (
????
????
)
?? +?? (
????
????
)
=
-(?? 2
-?? 2
)?? (
????
????
)
?? +?? (
????
????
)
Putting the above values of (?? 2
+?? ) and (?? 2
+?? ) in (i),
We have,
?? 2
[?? +?? (
????
????
)]
(?? 2
-?? 2
)?? -
?? 2
[?? +?? (
????
????
)]
(?? 2
-?? 2
)?? (
????
????
)
=1
or [?? +?? (
????
????
)][?? -?? (
?? ?? ????
)]=?? 2
-??
which is the differential equation of the family (i). Replacing ???? /???? by ( -?????????? ) in (ii),
the differential equation of the required orthogonal trajectories is
or [?? +?? (-
????
????
)][?? -?? (-
????
????
)]=?? 2
-?? 2
Read More
|
387 videos|203 docs
|
Related Exams
About this Document
|
Nov 21, 2024 Last updated |
Document Description: Formulation of Differential Equations for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Formulation of Differential Equations have been prepared according to the UPSC exam syllabus. Information about Formulation of Differential Equations covers topics
like and Formulation of Differential Equations Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Formulation of Differential Equations.
Introduction of Formulation of Differential Equations in English is available as part of our Mathematics Optional Notes for UPSC
for UPSC & Formulation of Differential Equations in Hindi for Mathematics Optional Notes for UPSC course.
Download more important topics related with notes, lectures and mock test series for UPSC
Exam by signing up for free. UPSC: Formulation of Differential Equations | Mathematics Optional Notes for UPSC
Description
Full syllabus notes, lecture & questions for Formulation of Differential Equations | Mathematics Optional Notes for UPSC - UPSC | Plus excerises question with solution to help you revise complete syllabus for Mathematics Optional Notes for UPSC | Best notes, free PDF download
Information about Formulation of Differential Equations
In this doc you can find the meaning of Formulation of Differential Equations defined & explained in the simplest way possible. Besides explaining types of
Formulation of Differential Equations theory, EduRev gives you an ample number of questions to practice Formulation of Differential Equations tests, examples and also practice UPSC
tests
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Formulation of Differential Equations | Mathematics Optional Notes for UPSC
,MCQs
,practice quizzes
,Viva Questions
,Important questions
,past year papers
,Semester Notes
,Previous Year Questions with Solutions
,video lectures
,shortcuts and tricks
,Exam
,Sample Paper
,Formulation of Differential Equations | Mathematics Optional Notes for UPSC
,mock tests for examination
,Extra Questions
,Summary
,study material
,Objective type Questions
,Free
,ppt
,Formulation of Differential Equations | Mathematics Optional Notes for UPSC
;
Additional Information about Formulation of Differential Equations for UPSC Preparation
Formulation of Differential Equations Free PDF Download
The Formulation of Differential Equations is an invaluable resource that delves deep into the core of the UPSC 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 Formulation of Differential Equations now and kickstart your journey towards success in the UPSC exam.
Importance of Formulation of Differential Equations
The importance of Formulation of Differential Equations cannot be overstated, especially for UPSC aspirants.
This document holds the key to success in the UPSC 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.
Formulation of Differential Equations Notes
Formulation of 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 Formulation of 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, Formulation of Differential Equations Notes on EduRev are your ultimate resource for success.
Formulation of Differential Equations UPSC Questions
The "Formulation of Differential Equations UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
UPSC 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 Formulation of Differential Equations on the App
Students of UPSC can study Formulation of Differential Equations alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Formulation of 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 Formulation of Differential Equations is prepared as per the latest UPSC syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup