# 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

- Homogeneous linear partial differential equations in the form
- 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