Chapter 2 1
Chapter 2 2
Chapter 2 3
Example
Chapter 2 4
Chapter 2 5
EXAMPLE
Chapter 2 6
Solution
Chapter 2 7
Chapter 2 8
Chapter 2 9
Method for solving First Order
Differential Equations
Methods
Variable SeparableVariable Separable
Reducible to variable separable Reducible to variable separable
Exact Differential EquationExact Differential Equation
Integrating FactorIntegrating Factor
Chapter 2
Separable Variable
x is independent variable and y is dependent variable
or
are separable forms of the differential equation
or
General solution can be solved by directly integrating both the sides
+ cWhere c is constant of integration
11DO YOU REMEMBER INTEGRATION FORMULA
Separation of Variables
and are separable
but is not separable.
xy xy y
y
x yy
x y
Definition A differential equation of the type y’ = f(x)g(y) is separable.
Example
x
yy
Example
Separable differential equations can often be solved with direct integration. This may lead to an equation which defines the solution implicitly rather than directly.
2 2
2 212 2
y xC y x C
ydy xdx
Chapter 2 13
EXAMPLE:
Chapter 2 14
EXAMPLE:
Chapter 2 15
To find the particular solution, we apply the given initial condition, when x =1, y = 3
is solution of initial value problem
Chapter 2 16
Chapter 2 17
Chapter 2 18
xdxdyyy 2ln2
xdxdyyy 2ln2
cxyyyy 22 ln
xdxdyyey y 22
xdxdyyey y 22
cxeyey yy 22
211 1sinsin yyyydy
xdxdyyy
xdxdyyy
2sin2
2sin21
1
cxyyyy 2212 1sin
Note1: If we have
Integrating by parts
Note.2. If we have
Integrating by parts
Note.3. If we have
yyyydy lnln
yyy eeydyye
Chapter 2 19
Chapter 2 20
Chapter 2 21
Method
Homogeneous EquationsReducible to separable
Chapter 2 23
Homogenous Differential Equations
A differential equation
Homogenous differential equation if
every t, where t R
isyxfdx
dy ),(
, , nf tx ty t f x y for
Chapter 2 24
Example:1. Show that differential equation is homogenous differential equation.
dxyxxydy 22
Solution: xy
yx
dx
dy 22
xy
yxyxf
22
,
xyt
ytxttytxf
2
2222
,
2 2 2 2 2
2,
t x y x yf x y
t xy xy
Differential equation is homogeneous
Differential equation is homogeneous
Chapter 2 25
METHOD for solving Homogenous differential equations
dx
duxu
dx
dy
uxy
xduudxdy
Substitute
Substitute
OR
vyx
ydvvdydx dy
dvyv
dy
dx
Chapter 2 26
Using substitution the homogeneous differential equation
is reduce to separable variable form.Example:2 Solve the homogenous differential
equation
xy
yx
dx
dy 22
Solution:Rewriting in the form : 0,, dyyxNdxyxM
.
022 xydydxyxuxy xduudxdy substitute and
Chapter 2 27
02222 xduudxuxdxuxx0322222 duuxdxxudxuxdxx
032 duuxdxx
duuxdxx 32
udux
dxx
3
2
udux
dx is variable separable form
udxx
dx cu
x 2
ln2
cx
yx
2
2
2
1ln is general solution.
Chapter 2 28
Note. Selection of substitution Differential Equation depends on
number of terms of coefficients yxandyxM ,N ,
01321 dydx uxy 1.
If , then take
2.
If
03211 dydx , then take x vx
3.
If 02121 dydx , then take
x = vy or y = ux
Chapter 2 29
Example:. Solve the Differential Equation by using appropriate substitution
0222 dyxdxxxyy Solution: Differential equation is homogeneous as degree of each term is same, hence we can use either y = ux or x = vy as substitution
xduudxdy
uxy
Let
Substituting y and dy in the given equation
duxudxxdxxdxuxdxxu
xduudxxdxxuxxu322222
22222
duxdxux
duxdxxdxxu322
3222
1
(1 / 2)
Chapter 2 30
.tanln
tanln
1
1
1
2
cx
yx
cux
u
du
x
dx
21 u
du
x
dx
is Separable form
Integrating both the sides
is general solution of the differential equation
Separating variable u and x (2 / 2)
Chapter 2 31
Example: Show that differential equation
22 943 yxdx
dyxy is homogeneous
dxyxxydy 22 943Solution:
xdvudxdyuxy ,
2 2 2
2 2 3 2 2 2
3 . 4 9
3 3 4 9
x ux udx xdu x u x dx
x u dx ux du x dx u x dx
dxuxdxxudxxduux 222223 64643
(1 / 2)
Chapter 2 32
x
dx
u
udu
264
3
x
dx
u
udu264
3
ududz
uz
12
64 2
Let
x
dx
z
dz
4
1
.ln64l4
1
lnln4
1
2
2
cxx
yn
cxz
is general solution of the differential equation
is Separable form
Integrating both the sides
(2 / 2)
Chapter 2 33
Hom
ogen
eous
Diff
eren
tial E
quati
on
Cha
pter
2
Chapter 2 34
(1 / 3)
Hom
ogen
eous
Diff
eren
tial E
quati
on
Cha
pter
2
Chapter 2 35
(2 / 3)
Hom
ogen
eous
Diff
eren
tial E
quati
on
Cha
pter
2
Chapter 2 36
(3 / 3)H
omog
eneo
us D
iffer
entia
l Equ
ation
C
hapt
er 2
Chapter 2 37
Hom
ogen
eous
Diff
eren
tial E
quati
on
Cha
pter
2
Chapter 2 38
(1 / 2)
Hom
ogen
eous
Diff
eren
tial E
quati
on
Cha
pter
2
Chapter 2 39is general solution of differential equation
(2 / 2)H
omog
eneo
us D
iffer
entia
l Equ
ation
C
hapt
er 2
Chapter 2 40
Diff
eren
tial E
quati
on
Cha
pter
2
Chapter 2 41
Diff
eren
tial E
quati
on
Cha
pter
2
Chapter 2 42
is general solution of differential equation
Diff
eren
tial E
quati
on
Cha
pter
2
