Transcript 02.4 Homogeneous

Method
Reducible
to
separable
Homogeneous Equations
Chapter 2
Homogeneous Differential Equation
Homogenous Differential Equations
A differential equation
dy
 f ( x, y ) is
dx
Homogenous differential equation if
every t, where t  R
f tx, ty   t n f  x, y  for
Chapter 2
2
Chapter 2
Homogeneous Differential Equation


xydy  x 2  y 2 dx
Example:1.
Show that differential equation
is homogenous differential equation.
dy x 2  y 2

dx
xy
Solution:
Differential equation is homogeneous
x2  y2
f x, y  
xy
t x t y
f tx, ty  
t 2 xy
2

2
2
2

t 2  x2  y 2 
t 2 xy
x2  y 2

 f  x, y 
xy
Differential equation is homogeneous
Chapter 2
3
Homogeneous Differential Equation
Chapter 2
METHOD for solving Homogenous differential equations
Substitute
y  ux
dy
du
ux
dx
dx
OR
Substitute
dy  udx  xdu
x  vy
dx
dv
v y
dy
dy
dx  vdy  ydv
Chapter 2
4
.
Homogeneous Differential Equation
Chapter 2
Using substitution the homogeneous differential equation
is reduce to separable variable form.
Example:2
Solve the homogenous differential equation
dy x 2  y 2

dx
xy
Solution:
Rewriting in the form :
x
substitute

M x, y dx  N x, y dy  0
 y dx  xydy  0
y  ux and dy  udx  xdu
2
2
Chapter 2
5

 x u dx  ux udx  xdu  0
2
2
Chapter 2
2
x 2 dx  x 2 u 2 dx  u 2 x 2 dx  ux 3 du  0
Homogeneous Differential Equation
x
2
x dx  ux du
x 2 dx  ux 3 du  0
2
3
2
x dx

udu
x3
dx
 udu
x
dx
 x   udx
is variable separable form
1  y2 
ln x   2   c
2  x Chapter 2
u2
ln x 
c
2
is general solution.
6
Chapter 2
Homogeneous Differential Equation
Note.
Selection of substitution Differential Equation depends on
M x, y  and Nx, y 
number of terms of coefficients
1.
If
1  2  3dx  1dy  0 ,
then take
y  ux
2.
If
1dx  1  2  3dy  0,
then take
x  vy
3.
If
1  2dx  1  2dy  0,
Chapter 2
then take
x = vy or y = ux
7
Solve the Differential Equation by using appropriate substitution
y
Chapter 2
Example:.
2

 xy  x dx  x dy  0
2
2
(1 / 2)
Homogeneous Differential Equation
Solution: Differential equation is homogeneous as degree of each term is same,
hence we can use either y = ux or x = vy as substitution
Let y  ux
dy  udx  xdu
Substituting y and dy in the given equation
u
2

x 2  ux 2  x 2 dx  x 2 udx  xdu  
u 2 x 2 dx  ux 2 dx  x 2 dx  x 2 udx  x 3 du  
u 2 x 2 dx  x 2 dx  x 3 du  


x 2 u 2  1 dx  x 3 du
Chapter 2
8
Homogeneous Differential Equation
Chapter 2
Separating variable u and x
(2 / 2)
dx
du

is Separable form
2
x 1 u
Integrating both the sides
dx
du
 x  1 u2
ln x  tan 1 u  c
 y
ln x  tan    c.
x
1
is general solution of the differential equation
Chapter 2
9
Homogeneous Differential Equation
Chapter 2
Example:
Show that differential equation
dy
3 xy
 4x 2  9 y 2
dx
Solution:

(1 / 2)
is homogeneous

3xydy  4 x 2  9 y 2 dx  
y  ux,
dy  udx  xdv


3 x.ux  udx  xdu   4 x 2  9u 2 x 2 dx 
3 x 2u 2 dx  3ux3du  4 x 2 dx  9u 2 x 2 dx 


3ux 3 du  4 x 2 dx  6u 2 x 2 dx  x 2 4  6u 2 dx
Chapter 2
10
Chapter 2
Homogeneous Differential Equation
3udu
dx

x
4  6u 2
is Separable form
(2 / 2)
Integrating both the sides
3udu
dx
 4  6u 2   x
Let
z  4  6u 2
dz  12udu
1 dz
dx


4 z
x
1
ln z  ln x  c
4
1 
y2 
ln 4  6 2   ln x  c.
4 
x 
is general solution of the differential equation
Chapter 2
11
Chapter 2
12
Homogeneous Differential Equation
Chapter 2
Homogeneous Differential Equation
Chapter 2
(1 / 3)
Chapter 2
13
Homogeneous Differential Equation
Chapter 2
(2 / 3)
Chapter 2
14
Homogeneous Differential Equation
Chapter 2
(3 / 3)
Chapter 2
15
Chapter 2
16
Homogeneous Differential Equation
Chapter 2
Homogeneous Differential Equation
Chapter 2
(1 / 2)
Chapter 2
17
Homogeneous Differential Equation
Chapter 2
(2 / 2)
is general solution Chapter
of differential
equation
2
18
Chapter 2
19
Differential Equation
Chapter 2
Chapter 2
20
Differential Equation
Chapter 2
Chapter 2
Differential Equation
is general solution of differential equation
Chapter 2
21
Differential Equation
Chapter 2
Example : By Substitution