# PARTIAL DIFFERENTIAL EQUATIONS

Partial differential equation are those equations which involves Partial derivatives.

• Notation: Let

## RULE FOR FORMATION OF PARTIAL DIFFERENTIAL EQUATIONS BY ELIMINATION OF ARBITRARY CONSTANTS

• If the number of arbitrary constants to be eliminated is equal to the number of independent variables we get a first order partial differential equation. If the number of arbitrary constants is more than the number of independent variables, we get a partial differential equations of higher order.

### RULE FOR FORMATION OF PARTIAL DIFFERENTIAL EQUATIONS BY ELIMINATION OF ARBITRARY FUNCTIONS

• If the number of arbitrary functions to be eliminated is equal to one then the required partial differential equations will be first order otherwise it will be of higher order.
• When an equation involves one arbitrary function the required partial differential equations always reduce to a Lagranges the partial differential equations of the form .
• Any solution which contains the same number of arbitrary constants as the number of independent variables is called a complete integral or solution.
• Any solution which is got by giving particular values to the arbitrary constants in a complete integral is called a particular integral.
• Let  Φ (x, y, z, a, b ) = 0   →( 1 ) be the complete integral of ƒ (x, y, z, p, q ) = 0 →( 2 ) where a and b are arbitrary constants.
• To eliminate a and b let us partial differentiate ( 1 ) . write equation a and b and equating it to zero, we get
•   and
•  from equations (1),  (3),  (4) eliminate  ‘ a ‘ and ‘ b ‘ we  obtain the singular integral of ( 2 )
• In the complete integral ( 1 )  put b = ƒ(a) then equations (1) becomes

Φ ( x, y, z, , ƒ(a) ) = 0  → ( 5 )

•   Now partially differentiating ( 1 )  write the equations we have

•  Eliminating between equations ( 5 ) and (6), we get general integral of ƒ ( x, y, z, p, q ) = 0

#### FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS

• An equation which involves partial derivatives   and  only is known as First order partial differential equations. The general first order partial differential equation is  ƒ ( x, y, z, p, q ) = 0 where and .
###### TYPE  (1) :   F(p ,q) = 0
• A partial differential equations which involves p and q only and the variables x, y, z do not  occur explicitly. Let  z = ax + by + c be the solution of the equations.
###### Type (2) CLAIRAUT’S FORM
• It is the form  z = px + qy + ƒ (p , q ). To find complete solution for clairaut’s type, replace p by a and q by b.
• Complete solution in the form  is z = ax + by + ƒ (a ,b ) where a, b are arbitrary constants.
###### Type (3) EQUATIONS OF TYPE F (z,p,q) = 0
• In these types of equations x and y do not appear explicitly. Let z =( fx + ay) be a trail solution. Let , substituting p and q in F (z,p,q) = 0 which is a first order ordinary differential equations on the integrating we get the complete solution.
###### Type (4)
• A first order partial differential equation is separable if it can be written as (x , p). Let (say)    Express x in  terms of p and K and y in terms of q and K . Substituting p and q in dz= pdx + qdy and integrating we get the complete integral . The singular and general integrals can be found out as usual.
###### Type (5) EQUATIONS REDUCIBLE TO THE STANDARD FORM

Case 1

• An equation of the form   where m and n are constants can be reduced to by the type (1) case by using the substituting and  , and .

Case 2

• An equation of the form  can be transformed to the type (3) by using the substitution and if ; and .

Case 3

• Put X = log x and Y = log y if m =1  and n = 1 in the above two cases.

Case 4

• An equation of the form where K is any constant can be transformed into the type (1) by proper substitution.                                If   put and if  put

Case 5 :

•  An equation of the form )=0 can be reduced into type ( 1 ) by putting  ,,, if  or by putting X = log x, Y = logy, Z = log z  if m =1, n =1 and K = -1.
##### LAGRANGE’ S LINEAR EQUATIONS
• A linear partial differential equation of first order is known as Lagrange’ s equations. Lagrange’ s linear equation is in the form  where P, Q and R are functions x, y, z.
• Elimination of the arbitrary function from the relation ( u , v )= 0 give us the lagrange’ s linear equation . We can find two functions  u , v such as that the eliminating of from  ( u , v )=0  Gives us the Langrange’ s linear partial differential equations , then  (u,v) =0 is the general solution where u and v are functions of x,y,z.

WORKING RULE FOR SOLVING LAGRANGE’ S PARTIAL DIFFERENTIAL EQUATIONS

• Put the partial differential equations of the first order in the form,
• Write the Lagrange’ s auxiliary equation
•   Solve (2) and let u (x,y,z)= and v(x,y,z)= be two independent solutions.
• The solution are written in the form ( u , v )=0 or u= (v) or v= (u).
###### METHOD OF GROUPING
• Given   we can solve it by variable separable method which gives us two independent solution u (x,y,z) = and v(x,y,z)= substituting u and v in ( u , v )=0 we get the equation for the Lagrange’s equation.
###### METHOD OF MULTIPLIERS
• Given the subsidiary equations   we can solve it as follows . We known that if =
• Where the two sets of multipliers  (l,m,n) , (l’, m’, n’) may be constants or functions of x,y,z. Choosing (l,m,n) such that lP +mQ+nR=0. We have l dx + m dy + n dz=0. Hence integrating we get u= u (x,y,z)= as one of the solutions similarly we have  l’dx + m’dy + n’dz=0. Which on integrating we give as v=v(x,y,z)= as the other solution the general  solution is (u , v)=0.
###### PARTIAL HOMOGENEOUS EQUATIONS OF HIGHER ORDER
• Linear partial differential equations of higher order with constants coefficients are
1.  Homogeneous linear partial differential equations in the form
2.  Non homogeneous linear differential equation is in the form
• If the roots of an arbitrary equation is real or complex and the different solution is
• If the roots are real and equal to the solution is
###### NON HOMOGENEOUS LINEAR PARTIAL DIFFERENTIAL EQUATIONS
• If the polynomial expression f(D , D’) is not homogeneous in f(D,D’) z = f (x,y) then the equation is known as the non homogeneous linear equation.
• Assume that where c,h,k are arbitrary constants then the solution is in f(D D’)z=0

c.f(h,k) f(h,k)=0

• If f(D D’) is of degree r in D’ then f(h,k)=0 will be the rth degree in k . solving for k from in terms of h we get where s =1,2,3,…….r
• Hence where s=1,2,…..r are separates solutions of f(D D’) z=0
• The arbitrary values of and h we get

to be the solution of f(D D’)z=0 . Hence the most general solution of