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UNIT-2
ORDINARY DIFFERENTIAL EQUATION
Introduction:In mathematics, a differential equation is an equation that relates one
or more functions and their derivatives
...
Such relations are common
...

Mainly the study of differential equations consists of the study of their
solutions (the set of functions that satisfy each equation)
...


DIFFERENTIAL EQUATION (DE)
An equation involving derivatives of one or more dependent variables
with respect to one or more independent variables is called a differential
equation
...
i
...
Otherwise we say that y does not
depend on x
...


(3)

Equation (1), (2) and (3) are called differential equations,
equation (4) is called simultaneous differential equation
...


(5)

In equation (2), Independent variables are 'x', 'y', 'z' and
dependent variable is 'u' which depends on x, y and z
...


ORDINARY DIFFERENTIAL EQUATION (ODE)
The equations having derivatives with respect to only one
independent variable are called ODE
...
 in



the differential

equation are called ordinary derivatives
...


PARTIAL DIFFERENTIAL EQUATION
The equations having derivatives with respect to at least two
independent variable are called ODE
...

x 2 y 2 z 2
 2u
u
5
 10
2
x
y

Note:(1)

  2 3

,
,

...

(2)

In other words, DE's having more than one independent
variables are called PDE
...

Example:d 4 x d 2 x dx


 et
dt 4 dt 2 dt
dx dy

 10
dt dt

   (1)
     (2)

2

dx dy
dx dy

 1, 3 
 0      (3)
dt dt
dt dt

d2x
 10 x  cos t
dt 2

   (4)

 2u  2 u  2u


 0
...

Since the highest order derivative in ODE (2) is 1, therefore the
order of the ODE is 1
...

Since the highest order derivative in ODE (4) is 2, therefore the
order of the ODE is 2
...


DEGREE OF A DIFFERENTIAL EQUATION
The degree of a DE is the power of the highest order derivative,
after the equation has been made rational and integral in all of its
derivatives
...
   (5)
 x   y   z 
3

  3u    u 
 3   5
  10
 x   y 

   (6)

Since the power of highest order derivative in ODE (1), (2) and
(3) are 1, therefore the degree of the ODE's are 1
...

Since the power of highest order derivative in PDE (5) is 2,
therefore the degree of the ODE's is 2
...

Since the power of highest order derivative in ODE (1) is 1,
therefore the degree of the ODE is 1
...

y x

Theorem
The Necessary and Sufficient condition for the DE Mdx  Ndy  0
to be exact if

M N
M N

 0 or


...

Solution:The given equation 2 xydx  ( x 2  3 y 2 )dy  0
...
1  2 x
y
y

-----(3)

Differentiate N partially with respect to 'x' (Assuming 'y' constant),
we get

N  ( x 2 )


 3 y 2 (1)  2 x  3 y 2
...

Hence the solution of given exact DE is given by


y constant

M x   (terms of N not containing x)dy=c

2 xy x   (3 y 2 )dy=c


y constant

2y







2

xx  3 ( y )dy=c

y constant

x n 1 
 x x  n  1 
n

 x 2   y3 
2 y    3   c
 2   3 

yx2  y3  c

Which is required solution
...

Solution:The given DE is (2 ye2 x  2 x cos y)dx  (e2 x  x 2 sin y)dy  0
...
1  2 x ( sin y )
y

M
 2e 2 x  2 x sin y
y





 cos ax

 a sin ax 
x


----(3)

Differentiate N partially with respect to x (Assuming 'y' constant),
we get
N


 e 2 x  x 2 sin y
x x
x

N


 e 2 x  sin y x 2
x x
x




N
 2e 2 x  sin y
...

Hence the solution of given exact DE is given by
M x   (terms of N not containing x)dy=c


y constant

(2 ye 2 x  2 x cos y )x   (0)dy=c


y constant

(2 ye2 x  2 x cos y )x  c


y constant

2y



e 2 x x  cos y

y constant

2 y
...
 c
...


Which is the required solution
...
-----(3)
Comparing the equations (2) and (3), we get
M  (3 x2  2 xy)

& N  (2 y  x 2 )

Differentiate M partially with respect to 'y' (Assuming 'x' constant),
we get
M


 3 x 2  2 xy
y y
y
M


 3 x 2 (1)  2 x y
y
y
y
M
 3 x 2
...
1
y
M
 2x
y

----(4)

Differentiate N partially with respect to 'x' (Assuming 'y' constant),
we get
N


 2 y  x2
x x
x
N


 2 y (1)  x 2
x
x
x
N
 2 y
...

Hence the solution of given exact DE is given by
M x   (terms of N not containing x)dy=c


y constant

(3x 2  2 x y )x   (2 y )dy=c


y constant

3



x 2 x  2 y

y constant

3

x x  2 ydy=c


y constant

x3
x2
y2
 2 y
...
 c
3
2
2

x3  x 2 y  y 2  c
...
y e x dx  (2 y  e x )dy  0
2
...
( x 2  y 2  a 2 ) xdx  ( x 2  y 2  b2 ) ydy  0
4
...
( y 2 e xy  4 x 3 )dx  (2 xye xy  3 y 2 )dy  0

Ans
...
 y  1    cos y  dx  ( x  log x  x sin y )dy  0

Ans
...
(1  2 xy cos x2  2 xy)dx  (sin x 2  x2 )dy  0

Ans
...


x



dy y cos x  sin y  y
+
=0
dx sin x  x cos y  x

9
...
(3x 2 +6xy2 )dx+(6x 2 y+4y3 )dy=0

Ans
...
x 2  y 2  3  c( x2  y 2  1)5



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