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 .

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

)1......(0),,,,(),,,,(

qpzyxF

y

z

x

zzyxF

1) COMPLETE INTEGRAL SOLUTION

2) PARTICULAR SOLUTION

3) SINGULAR SOLUTION

4) GENERAL SOLUTION

Let

be the Partial Differential Equation.

The complete integral of equation (1) is given by

where a and b are two arbitrary constants

)1......(0),,,,(),,,,(

qpzyxF

y

z

x

zzyxF

)2(..........0),,,,( bazyx

A solution obtained by giving the particular

values to the arbitrary constants in a complete

integral is called particular solution.

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

0,0

0),,,,(

ba

bazyx

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

)2(..........0),,,,( bazyx

)(afb

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))(,,,,( afazyx

)4(..........0)(

af

ba

TYPE-1

The Partial Differential equation of the form

has solution

and

0),( qpf

cbyaxz 0),( baf

TYPE-2The partial differentiation equation of the form

is called Clairaut’s form of partial differential

equations.

),( bafbyaxz

TYPE-3 If the partial differential equations is given

by

Then assume that

0),,( qpzf

)(

)(

uz

ayxu

ayxz

du

dzaa

u

z

y

u

u

z

y

zq

du

dz

u

z

x

u

u

z

x

zp

.

1.

TYPE-4 The partial differential equation of the given

form can be solved by assuming

),(),(

),(),(

),(),(

ayqaqyg

axpapxf

aqygpxf

dyaydxaxdz

dyy

zdx

x

zdz

),(),(

1.Solve the pde and find the complete and singular solutions

Solution

Complete solution is given by

with

12 qp

cbyaxz

1

1

2

2

ab

ba

cyaaxz )1( 2

d.w.r.to. a and c then

01

2

c

z

ayxa

z

Which is not possible

Hence there is no singular solution

2.Solve pde

(or) xyy

z

x

z

xypq

))((

Solutionq

y

x

p

Assume that

dya

yaxdxqdypdxdz

a

yqaxp

aq

y

x

p

,

Integrating on both sides

ba

yxaz

22

22

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