......

1. PARTIAL DIFFERENTIAL
EQUATIONS

2.  It is defined as an equation involving two or
more independent variables like x,y……., a
dependent variable like u and its partial
derivatives.
 Partial Differential Equation can be formed
either by elimination of arbitrary constants or
by the elimination of arbitrary functions from
a relation involving three or more variables .
FORMATION OF PARTIAL DIFFERENTIAL
EQUATIONS

3.  The general form of a first order partial
differential equation is
where x, y are two independent variables, z
is the dependent variable and p = zx and
q = zy
GENERAL FORM
)
1
......(
0
)
,
,
,
,
(
)
,
,
,
,
( 





q
p
z
y
x
F
y
z
x
z
z
y
x
F

4. 1) COMPLETE INTEGRAL SOLUTION
2) PARTICULAR SOLUTION
3) SINGULAR SOLUTION
4) GENERAL SOLUTION
DIFFERENT INTEGRALS OF PARTIAL
DIFFERNETIAL EQUATION

5.  Let
be the Partial Differential Equation.
 The complete integral of equation (1) is given
by
 where a and b are two arbitrary constants
COMPLETE SOLUTION
)
1
......(
0
)
,
,
,
,
(
)
,
,
,
,
( 





q
p
z
y
x
F
y
z
x
z
z
y
x
F
)
2
(
..........
0
)
,
,
,
,
( 
b
a
z
y
x


6.  A solution obtained by giving the particular
values to the arbitrary constants in a complete
integral is called particular solution.
PARTICULAR SOLUTION

7.  It is the relation between those specific
variables which involves no arbitrary constant
and is not obtainable as a particular integral
from the complete integral.
 So, equation is
SINGULAR SOLUTION
0
,
0
0
)
,
,
,
,
(







b
a
b
a
z
y
x




8.  A relation between the variables involving
two independent functions of the given
variables together with an arbitrary function
of these variables is a general solution.
 In this given equation
assume an arbitrary relation of form
GENERAL SOLUTION
)
2
(
..........
0
)
,
,
,
,
( 
b
a
z
y
x

)
(a
f
b 

9.  So, our earlier equation becomes
 Now, differentiating (2) with respect to a and
thus we get,
 If the eliminator of (3) and (4) exists, then it is
known as general solution.
)
3
.........(
0
))
(
,
,
,
,
( 
a
f
a
z
y
x

)
4
(
..........
0
)
( 






a
f
b
a



10.  TYPE-1
The Partial Differential equation of the form
has solution
and
STANDARD TYPES OF FIRST ORDER
EQUATIONS
0
)
,
( 
q
p
f
c
by
ax
z 

 0
)
,
( 
b
a
f

11.  TYPE-2
The partial differentiation equation of the form
is called Clairaut’s form of partial differential
equations.
)
,
( b
a
f
by
ax
z 



12.  TYPE-3
 If the partial differential equations is given by
 Then assume that
0
)
,
,
( 
q
p
z
f
)
(
)
(
u
z
ay
x
u
ay
x
z








13. du
dz
a
a
u
z
y
u
u
z
y
z
q
du
dz
u
z
x
u
u
z
x
z
p
























.
1
.

14.  TYPE-4
 The partial differential equation of the given
form can be solved by assuming
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
a
y
q
a
q
y
g
a
x
p
a
p
x
f
a
q
y
g
p
x
f










dy
a
y
dx
a
x
dz
dy
y
z
dx
x
z
dz
)
,
(
)
,
( 










15.  1.Solve the pde and find the
complete and singular solutions
 Solution
Complete solution is given by
 with
EXAMPLES
1
2

 q
p
c
by
ax
z 


1
1
2
2





a
b
b
a

16. c
y
a
ax
z 


 )
1
( 2
d.w.r.to. a and c then
0
1
2








c
z
ay
x
a
z
Which is not possible
Hence there is no singular solution

17.  2.Solve pde
(or) xy
y
z
x
z
xy
pq






)
)(
(
Solution q
y
x
p

Assume that
dy
a
y
axdx
qdy
pdx
dz
a
y
q
ax
p
a
q
y
x
p









,

18.  Integrating on both sides
b
a
y
x
a
z 


2
2
2
2

19. THANK YOU
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BY- CHIRAYU GAUTAM