Electrical Engineering (EE) Exam > Electrical Engineering (EE) Notes > Short Notes for Electrical Engineering > Short Notes: Differential Equations
Short Notes: Differential Equations | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
DIFFERENTIAL EQUATIONS
? The order of a deferential equation is the order of highest derivative appearing in it.
? The degree of a differential equation is the degree of the highest derivative occurring in it,
after the differential equation is expressed in a form free from radicals & fractions.
For equations of first order & first degree
? Variable Separation method
Collect all function of x & dx on one side.
Collect all function of y & dy on other side.
like f(x) dx = g(y) dy
solution: ? ? ? ? f x dx g y dy c ??
??
? Exact differential equation
An equation of the form
M(x, y) dx + N (x, y) dy = 0
For equation to be exact.
MN
yx
??
?
??
; then only this method can be applied.
The solution is
a = ?
??
M dx (termsofNnotcontainingx)dy
? Integrating factors
An equation of the form
P(x, y) dx + Q (x, y) dy = 0
This can be reduced to exact form by multiplying both sides by IF.
If
?? ??
?
??
??
??
1 P Q
Q y x
is a function of x, then
R(x) =
?? ??
??
??
??
??
1 P Q
Q y x
Integrating Factor
IF = exp ? ?
? ?
R x dx
?
Otherwise, if
QP
1
P
xy
?? ??
?
??
??
??
is a function of y
Page 2
DIFFERENTIAL EQUATIONS
? The order of a deferential equation is the order of highest derivative appearing in it.
? The degree of a differential equation is the degree of the highest derivative occurring in it,
after the differential equation is expressed in a form free from radicals & fractions.
For equations of first order & first degree
? Variable Separation method
Collect all function of x & dx on one side.
Collect all function of y & dy on other side.
like f(x) dx = g(y) dy
solution: ? ? ? ? f x dx g y dy c ??
??
? Exact differential equation
An equation of the form
M(x, y) dx + N (x, y) dy = 0
For equation to be exact.
MN
yx
??
?
??
; then only this method can be applied.
The solution is
a = ?
??
M dx (termsofNnotcontainingx)dy
? Integrating factors
An equation of the form
P(x, y) dx + Q (x, y) dy = 0
This can be reduced to exact form by multiplying both sides by IF.
If
?? ??
?
??
??
??
1 P Q
Q y x
is a function of x, then
R(x) =
?? ??
??
??
??
??
1 P Q
Q y x
Integrating Factor
IF = exp ? ?
? ?
R x dx
?
Otherwise, if
QP
1
P
xy
?? ??
?
??
??
??
is a function of y
S(y) =
QP
1
P
xy
?? ??
?
??
??
??
Integrating factor, IF = exp
? ? ? ?
?
S y dy
? Linear Differential Equations
An equation is linear if it can be written as:
? ? ? ?
?? y' P x y r x
If r(x) = 0 ; equation is homogenous
else r(x) ? 0 ; equation is non-homogeneous
y(x) =
? ? p x dx
ce
?
?
is the solution for homogenous form
for non-homogenous form, h = ? ? P x dx
?
? ?
h h
y x e e rdx c
?
??
??
??
?
? Bernoulli’s equation
The equation ??
n
dy
Py Qy
dx
Where P & Q are function of x
Divide both sides of the equation by
n
y & put
? ? 1 n
yz
?
?
? ? ? ?
? ? ? ?
dz
P 1 n z Q 1 n
dx
This is a linear equation & can be solved easily.
? Clairaut’s equation
An equation of the form y = Px + f (P), is known as Clairaut’s equation where P =
? ?
dy
dx
The solution of this equation is
y = cx + f (c) where c = constant
Linear Differential Equation of Higher Order
Constant coefficient differential equation
n
n n1
n1
n
1
d y d y
k .............. k y X
dx dx
?
?
? ? ? ?
Page 3
DIFFERENTIAL EQUATIONS
? The order of a deferential equation is the order of highest derivative appearing in it.
? The degree of a differential equation is the degree of the highest derivative occurring in it,
after the differential equation is expressed in a form free from radicals & fractions.
For equations of first order & first degree
? Variable Separation method
Collect all function of x & dx on one side.
Collect all function of y & dy on other side.
like f(x) dx = g(y) dy
solution: ? ? ? ? f x dx g y dy c ??
??
? Exact differential equation
An equation of the form
M(x, y) dx + N (x, y) dy = 0
For equation to be exact.
MN
yx
??
?
??
; then only this method can be applied.
The solution is
a = ?
??
M dx (termsofNnotcontainingx)dy
? Integrating factors
An equation of the form
P(x, y) dx + Q (x, y) dy = 0
This can be reduced to exact form by multiplying both sides by IF.
If
?? ??
?
??
??
??
1 P Q
Q y x
is a function of x, then
R(x) =
?? ??
??
??
??
??
1 P Q
Q y x
Integrating Factor
IF = exp ? ?
? ?
R x dx
?
Otherwise, if
QP
1
P
xy
?? ??
?
??
??
??
is a function of y
S(y) =
QP
1
P
xy
?? ??
?
??
??
??
Integrating factor, IF = exp
? ? ? ?
?
S y dy
? Linear Differential Equations
An equation is linear if it can be written as:
? ? ? ?
?? y' P x y r x
If r(x) = 0 ; equation is homogenous
else r(x) ? 0 ; equation is non-homogeneous
y(x) =
? ? p x dx
ce
?
?
is the solution for homogenous form
for non-homogenous form, h = ? ? P x dx
?
? ?
h h
y x e e rdx c
?
??
??
??
?
? Bernoulli’s equation
The equation ??
n
dy
Py Qy
dx
Where P & Q are function of x
Divide both sides of the equation by
n
y & put
? ? 1 n
yz
?
?
? ? ? ?
? ? ? ?
dz
P 1 n z Q 1 n
dx
This is a linear equation & can be solved easily.
? Clairaut’s equation
An equation of the form y = Px + f (P), is known as Clairaut’s equation where P =
? ?
dy
dx
The solution of this equation is
y = cx + f (c) where c = constant
Linear Differential Equation of Higher Order
Constant coefficient differential equation
n
n n1
n1
n
1
d y d y
k .............. k y X
dx dx
?
?
? ? ? ?
Where X is a function of x only
a. If
n 12
y , y ,.........., y are n independent solution, then
nn 1 1 2 2
c y c y .......... c y x ? ? ? ? is complete solution
where
1 2 n
c ,c ,..........,c are arbitrary constants.
b. The procedure of finding solution of n
th
order differential equation involves
computing complementary function (C. F) and particular Integral (P. I).
c. Complementary function is solution of
n
n n1
n1
n
1
d y d y
k ............. k y 0
dx dx
?
?
? ? ? ?
d. Particular integral is particular solution of
n
n n1
n1
n
1
d y d y
k ............ k y x
dx dx
?
?
? ? ? ?
e. y = CF + PI is complete solution
Finding complementary function
? Method of differential operator
Replace
d
dx
by D ?
dy
Dy
dx
?
Similarly
n
n
d
dx
by
n
D ?
n
n
n
dy
Dy
dx
?
n
n n1
n1
n
1
d y d y
k ............ k y 0
dx dx
?
?
? ? ? ? becomes
? ?
n n 1
n
1
D k D ........... k y 0
?
? ? ? ?
Let
12 n
m ,m ,............,m be roots of
n
1
n1
n
D k D ................ K 0
?
? ? ? ………….(i)
Page 4
DIFFERENTIAL EQUATIONS
? The order of a deferential equation is the order of highest derivative appearing in it.
? The degree of a differential equation is the degree of the highest derivative occurring in it,
after the differential equation is expressed in a form free from radicals & fractions.
For equations of first order & first degree
? Variable Separation method
Collect all function of x & dx on one side.
Collect all function of y & dy on other side.
like f(x) dx = g(y) dy
solution: ? ? ? ? f x dx g y dy c ??
??
? Exact differential equation
An equation of the form
M(x, y) dx + N (x, y) dy = 0
For equation to be exact.
MN
yx
??
?
??
; then only this method can be applied.
The solution is
a = ?
??
M dx (termsofNnotcontainingx)dy
? Integrating factors
An equation of the form
P(x, y) dx + Q (x, y) dy = 0
This can be reduced to exact form by multiplying both sides by IF.
If
?? ??
?
??
??
??
1 P Q
Q y x
is a function of x, then
R(x) =
?? ??
??
??
??
??
1 P Q
Q y x
Integrating Factor
IF = exp ? ?
? ?
R x dx
?
Otherwise, if
QP
1
P
xy
?? ??
?
??
??
??
is a function of y
S(y) =
QP
1
P
xy
?? ??
?
??
??
??
Integrating factor, IF = exp
? ? ? ?
?
S y dy
? Linear Differential Equations
An equation is linear if it can be written as:
? ? ? ?
?? y' P x y r x
If r(x) = 0 ; equation is homogenous
else r(x) ? 0 ; equation is non-homogeneous
y(x) =
? ? p x dx
ce
?
?
is the solution for homogenous form
for non-homogenous form, h = ? ? P x dx
?
? ?
h h
y x e e rdx c
?
??
??
??
?
? Bernoulli’s equation
The equation ??
n
dy
Py Qy
dx
Where P & Q are function of x
Divide both sides of the equation by
n
y & put
? ? 1 n
yz
?
?
? ? ? ?
? ? ? ?
dz
P 1 n z Q 1 n
dx
This is a linear equation & can be solved easily.
? Clairaut’s equation
An equation of the form y = Px + f (P), is known as Clairaut’s equation where P =
? ?
dy
dx
The solution of this equation is
y = cx + f (c) where c = constant
Linear Differential Equation of Higher Order
Constant coefficient differential equation
n
n n1
n1
n
1
d y d y
k .............. k y X
dx dx
?
?
? ? ? ?
Where X is a function of x only
a. If
n 12
y , y ,.........., y are n independent solution, then
nn 1 1 2 2
c y c y .......... c y x ? ? ? ? is complete solution
where
1 2 n
c ,c ,..........,c are arbitrary constants.
b. The procedure of finding solution of n
th
order differential equation involves
computing complementary function (C. F) and particular Integral (P. I).
c. Complementary function is solution of
n
n n1
n1
n
1
d y d y
k ............. k y 0
dx dx
?
?
? ? ? ?
d. Particular integral is particular solution of
n
n n1
n1
n
1
d y d y
k ............ k y x
dx dx
?
?
? ? ? ?
e. y = CF + PI is complete solution
Finding complementary function
? Method of differential operator
Replace
d
dx
by D ?
dy
Dy
dx
?
Similarly
n
n
d
dx
by
n
D ?
n
n
n
dy
Dy
dx
?
n
n n1
n1
n
1
d y d y
k ............ k y 0
dx dx
?
?
? ? ? ? becomes
? ?
n n 1
n
1
D k D ........... k y 0
?
? ? ? ?
Let
12 n
m ,m ,............,m be roots of
n
1
n1
n
D k D ................ K 0
?
? ? ? ………….(i)
Case I: All roots are real & distinct
? ? ? ? ? ?
n 12
D m D m ............ D m 0 ? ? ? ? is equivalent to (i)
y =
1 2n
mx mx mx
n
12
c e c e ........... c e ? ? ?
is solution of differential equation
Case II: If two roots are real & equal
i.e.,
12
m m m ??
y =
? ?
2
3 n
mx
mx mx
n
13
c c x e c e .......... c e ? ? ? ?
Case III: If two roots are complex conjugate
1
m j ? ? ? ? ;
2
m j ? ? ? ?
y =
?
?? ? ? ? ? ?
?? 12
n
mx
n
x
e c 'cos x c 'sin x .......... c e
Finding particular integral
Suppose differential equation is
n n1
n
n 1
n1
d y d y
k .......... k y X
dx
dx
?
?
? ? ? ?
Particular Integral
PI =
? ?
? ?
? ?
? ?
? ?
? ?
? ? ? ? ? ?
? ? ?
n 12
n 12
W x W x W x
y dx y dx .......... y dx
W x W x W x
Where
n
12
y ,y ,............y are solutions of Homogenous from of differential equations.
? ?
? ? ? ? ? ?
n
12
n
12
nn n
n
12
y y y
y ' y ' y '
Wx
y y y
?
?
?
?
?
? ?
? ? ? ? ? ? ? ?
n 12 i1
n 12 i1
i
n n n n
n 12 i1
y y y0y
y ' y ' y '0 y '
Wx
0
y y y 1y
?
?
?
? ?
?
?
?
?
? ?
i
Wx is obtained from W(x) by replacing i
th
column by all zeroes & last 1.
Page 5
DIFFERENTIAL EQUATIONS
? The order of a deferential equation is the order of highest derivative appearing in it.
? The degree of a differential equation is the degree of the highest derivative occurring in it,
after the differential equation is expressed in a form free from radicals & fractions.
For equations of first order & first degree
? Variable Separation method
Collect all function of x & dx on one side.
Collect all function of y & dy on other side.
like f(x) dx = g(y) dy
solution: ? ? ? ? f x dx g y dy c ??
??
? Exact differential equation
An equation of the form
M(x, y) dx + N (x, y) dy = 0
For equation to be exact.
MN
yx
??
?
??
; then only this method can be applied.
The solution is
a = ?
??
M dx (termsofNnotcontainingx)dy
? Integrating factors
An equation of the form
P(x, y) dx + Q (x, y) dy = 0
This can be reduced to exact form by multiplying both sides by IF.
If
?? ??
?
??
??
??
1 P Q
Q y x
is a function of x, then
R(x) =
?? ??
??
??
??
??
1 P Q
Q y x
Integrating Factor
IF = exp ? ?
? ?
R x dx
?
Otherwise, if
QP
1
P
xy
?? ??
?
??
??
??
is a function of y
S(y) =
QP
1
P
xy
?? ??
?
??
??
??
Integrating factor, IF = exp
? ? ? ?
?
S y dy
? Linear Differential Equations
An equation is linear if it can be written as:
? ? ? ?
?? y' P x y r x
If r(x) = 0 ; equation is homogenous
else r(x) ? 0 ; equation is non-homogeneous
y(x) =
? ? p x dx
ce
?
?
is the solution for homogenous form
for non-homogenous form, h = ? ? P x dx
?
? ?
h h
y x e e rdx c
?
??
??
??
?
? Bernoulli’s equation
The equation ??
n
dy
Py Qy
dx
Where P & Q are function of x
Divide both sides of the equation by
n
y & put
? ? 1 n
yz
?
?
? ? ? ?
? ? ? ?
dz
P 1 n z Q 1 n
dx
This is a linear equation & can be solved easily.
? Clairaut’s equation
An equation of the form y = Px + f (P), is known as Clairaut’s equation where P =
? ?
dy
dx
The solution of this equation is
y = cx + f (c) where c = constant
Linear Differential Equation of Higher Order
Constant coefficient differential equation
n
n n1
n1
n
1
d y d y
k .............. k y X
dx dx
?
?
? ? ? ?
Where X is a function of x only
a. If
n 12
y , y ,.........., y are n independent solution, then
nn 1 1 2 2
c y c y .......... c y x ? ? ? ? is complete solution
where
1 2 n
c ,c ,..........,c are arbitrary constants.
b. The procedure of finding solution of n
th
order differential equation involves
computing complementary function (C. F) and particular Integral (P. I).
c. Complementary function is solution of
n
n n1
n1
n
1
d y d y
k ............. k y 0
dx dx
?
?
? ? ? ?
d. Particular integral is particular solution of
n
n n1
n1
n
1
d y d y
k ............ k y x
dx dx
?
?
? ? ? ?
e. y = CF + PI is complete solution
Finding complementary function
? Method of differential operator
Replace
d
dx
by D ?
dy
Dy
dx
?
Similarly
n
n
d
dx
by
n
D ?
n
n
n
dy
Dy
dx
?
n
n n1
n1
n
1
d y d y
k ............ k y 0
dx dx
?
?
? ? ? ? becomes
? ?
n n 1
n
1
D k D ........... k y 0
?
? ? ? ?
Let
12 n
m ,m ,............,m be roots of
n
1
n1
n
D k D ................ K 0
?
? ? ? ………….(i)
Case I: All roots are real & distinct
? ? ? ? ? ?
n 12
D m D m ............ D m 0 ? ? ? ? is equivalent to (i)
y =
1 2n
mx mx mx
n
12
c e c e ........... c e ? ? ?
is solution of differential equation
Case II: If two roots are real & equal
i.e.,
12
m m m ??
y =
? ?
2
3 n
mx
mx mx
n
13
c c x e c e .......... c e ? ? ? ?
Case III: If two roots are complex conjugate
1
m j ? ? ? ? ;
2
m j ? ? ? ?
y =
?
?? ? ? ? ? ?
?? 12
n
mx
n
x
e c 'cos x c 'sin x .......... c e
Finding particular integral
Suppose differential equation is
n n1
n
n 1
n1
d y d y
k .......... k y X
dx
dx
?
?
? ? ? ?
Particular Integral
PI =
? ?
? ?
? ?
? ?
? ?
? ?
? ? ? ? ? ?
? ? ?
n 12
n 12
W x W x W x
y dx y dx .......... y dx
W x W x W x
Where
n
12
y ,y ,............y are solutions of Homogenous from of differential equations.
? ?
? ? ? ? ? ?
n
12
n
12
nn n
n
12
y y y
y ' y ' y '
Wx
y y y
?
?
?
?
?
? ?
? ? ? ? ? ? ? ?
n 12 i1
n 12 i1
i
n n n n
n 12 i1
y y y0y
y ' y ' y '0 y '
Wx
0
y y y 1y
?
?
?
? ?
?
?
?
?
? ?
i
Wx is obtained from W(x) by replacing i
th
column by all zeroes & last 1.
Euler-Cauchy Equation
An equation of the form
n n 1
n n 1
n
n 1
n1
d y d y
x k x .......... k y 0
dx
dx
?
?
?
? ? ? ?
is called as Euler-Cauchy theorem
Substitute y = x
m
The equation becomes
? ? ? ? ? ?
??
? ? ? ? ? ? ? ? ?
??
m
n 1
m m 1 ........ m n k m(m 1)......... m n 1 ............. k x 0
The roots of equation are
Case I: All roots are real & distinct
1 2 n
mm
m
n
12
y c x c x ........... c x ? ? ? ?
Case II: Two roots are real & equal
12
m m m ??
? ?
3
m
m m
n
1 2 3
n
y c c nx x c x ........ c x ? ? ? ? ?
Case III: Two roots are complex conjugate of each other
1
mj ? ? ? ? ;
2
mj ? ? ? ?
y =
? ? ? ?
?
??
? ? ? ? ? ? ?
??
3
m
n
m
n 3
x Acos nx Bsin nx c x ........... c x
Read More
About this Document
|
4.67/5 Rating |
|
Dec 22, 2024 Last updated |
Document Description: Short Notes: Differential Equations for Electrical Engineering (EE) 2024 is part of Short Notes for Electrical Engineering preparation.
The notes and questions for Short Notes: Differential Equations have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Short Notes: Differential Equations covers topics
like and Short Notes: Differential Equations Example, for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Short Notes: Differential Equations.
Introduction of Short Notes: Differential Equations in English is available as part of our Short Notes for Electrical Engineering
for Electrical Engineering (EE) & Short Notes: Differential Equations in Hindi for Short Notes for Electrical Engineering course.
Download more important topics related with notes, lectures and mock test series for Electrical Engineering (EE)
Exam by signing up for free. Electrical Engineering (EE): Short Notes: Differential Equations | Short Notes for Electrical Engineering - Electrical Engineering (EE)
Description
Full syllabus notes, lecture & questions for Short Notes: Differential Equations | Short Notes for Electrical Engineering - Electrical Engineering (EE) - Electrical Engineering (EE) | Plus excerises question with solution to help you revise complete syllabus for Short Notes for Electrical Engineering | Best notes, free PDF download
Information about Short Notes: Differential Equations
In this doc you can find the meaning of Short Notes: Differential Equations defined & explained in the simplest way possible. Besides explaining types of
Short Notes: Differential Equations theory, EduRev gives you an ample number of questions to practice Short Notes: Differential Equations tests, examples and also practice Electrical Engineering (EE)
tests
|
Explore Courses for Electrical Engineering (EE) exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
practice quizzes
,Free
,past year papers
,Objective type Questions
,ppt
,Summary
,shortcuts and tricks
,Sample Paper
,Exam
,Short Notes: Differential Equations | Short Notes for Electrical Engineering - Electrical Engineering (EE)
,mock tests for examination
,Previous Year Questions with Solutions
,Short Notes: Differential Equations | Short Notes for Electrical Engineering - Electrical Engineering (EE)
,video lectures
,Extra Questions
,Important questions
,Short Notes: Differential Equations | Short Notes for Electrical Engineering - Electrical Engineering (EE)
,Semester Notes
,Viva Questions
,MCQs
,study material
;
Additional Information about Short Notes: Differential Equations for Electrical Engineering (EE) Preparation
Short Notes: Differential Equations Free PDF Download
The Short Notes: Differential Equations is an invaluable resource that delves deep into the core of the Electrical Engineering (EE) 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 Short Notes: Differential Equations now and kickstart your journey towards success in the Electrical Engineering (EE) exam.
Importance of Short Notes: Differential Equations
The importance of Short Notes: Differential Equations cannot be overstated, especially for Electrical Engineering (EE) aspirants.
This document holds the key to success in the Electrical Engineering (EE) 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.
Short Notes: Differential Equations
Short Notes: 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 Short Notes: 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, Short Notes: Differential Equations Notes on EduRev are your ultimate resource for success.
Short Notes: Differential Equations Electrical Engineering (EE) Questions
The "Short Notes: Differential Equations Electrical Engineering (EE) Questions" guide is a valuable resource for all aspiring students preparing for the
Electrical Engineering (EE) 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 Short Notes: Differential Equations on the App
Students of Electrical Engineering (EE) can study Short Notes: Differential Equations alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Short Notes: 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 Short Notes: Differential Equations is prepared as per the latest Electrical Engineering (EE) 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