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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
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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.
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