Module 3C

Partial Differential equation (First order)

3.1 First Order Partial differential equation:-

Partial differential equation are those equation which contain partial differential coefficient, independent variables and dependent variables. The independent variable will be denoted by x and y and the dependent variable by z . The partial differential equation coefficient are denoted as follows.

Orders

Order of a partial differential equation is the same as that of the order of the highest differential coefficient in it.

Method of forming partial differential equation

A partial differential equation is formed by two method

(i)                 By eliminating arbitrary constant

(ii)               By eliminating arbitrary function.

Method of elimination of arbitrary constant

Example. Form a partial differential equation from

Ans. Given
 

Equation contains two arbitrary constant a and c.

Differentiating equation 1 partially with respect to x we get

Differentiating equation 1 partially with respect to y we get

2y+2(z -c)   =0

y+(z -c) q=0     (3)

Eliminate c from (2) and (3) we get from (2) ( z -c)=

Putting the value of  (z -c) in (3) we get

Method of elimination of arbitrary function

Example: Form the partial differential equation from

Ans. Given

Differentiating equation 1 with respect to X and y we get

Dividing equation 2 by 3 we get

Solve the partial differential equation

Ans. Given that,

Differentiating (1) partially with respect to x and y we get

Noe eliminating a,b from these above equations we have

Which is required partial differential equation?

Que. Form a partial differential equation by eliminating a, b from

Ans. Given          (1)

Differentiating (1) partially with respect to x and y we get

Differentiating (i) and (ii) we have

3.2 Solution of first order linear and nonlinear partial differential equation

Lagrange's linear equation:-

Working rule:

Step1:- Write down the auxiliary equation

Step2:- Solve the above auxiliary equation

Let us two solution be

Step3:- Then is the required solution of

Example:-

Solve

Ans. AE are

From first two members of the equation

On integrating

From last two members of the equation

On integrating

From (2) and (3)

Method of multipliers

Let the AE be

l, m, n may be constant or function of x y z then we have

l, m, n are chosen in such a way that l p +m q + n R=0

Thus,

Solve this differential equation if the solution is

Similarly choose another set of multipliers and if the second solution is

Required solution is

Solve

Ans.

Using multipliers xyz we get

Each fraction=

On integrating

Again using l,m,n of multipliers

On integrating

From (1) and (2) we get

Partial differential equations Non linear in P and q

Type I

Equation of the type

Example: Solve

Ans. Let z=ax+by+c     (1)

Putting p=a , q=b in equation 2

Putting the value of b in equation 1

Type 2:- Equation of the type its solution is z=ax +by + f(a ,b)

Solve

Ans. Given,

Put p=a, q=b

Solution is

Solve. y2 zp-x2 zq=x2 y

Ans. The given equation is of the Lagrange’s form

Where,

The AE is

From

On integrating,

Also,

On integrating

General solution is

Solve.

Ans. AE is

On integrating

On integrating,

General solution is

Method of Multipliers

Solve. (a)

This is Lagrange’s equation

The subsidiary equation are

Using multipliers x, y, -1 we get

Each ratio =

On integrating we get

Each ratio =

On integrating we get

The general solutions are

Solve.  

Ans. Given,

The Lagrange’s subsidiary equation are

Choosing x, -y, -z as multipliers we get

On integrating

is an arbitrary constant.

Again choosing, y, x, -z as multipliers

On integrating,

is an arbitrary constant.

Hence required solution is

Partial Differential Equation non linear in p and q

Type A

Solve.

Ans. Here

The solution to given equation is

Where,

Hence b can be expressed in terms of a as

Thus complete solution is

Solve.

Ans. Here, F(pq)=pq+p+q

A complete solution of a given equation is

Since,

i.e.

The solution is

Type 2

Solve.

Ans. The given equation can be written as

Complete solution is

Solve.

Ans. The given equation is of the form

Its solution is given by

TEXTBOOKS/REFERENCES:

1. S. J. FARLOW, PARTIAL DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS, DOVER PUBLICATIONS, 1993.
2. R. HABERMAN, ELEMENTARY APPLIED PARTIAL DIFFERENTIAL EQUATIONS WITH FOURIER SERIES AND BOUNDARY VALUE PROBLEM, 4TH ED., PRENTICE HALL, 1998.
3. IAN SNEDDON, ELEMENTS OF PARTIAL DIFFERENTIAL EQUATIONS, MCGRAW HILL, 1964.
4. MANISH GOYAL AND N.P. BALI, TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS, UNIVERSITY SCIENCE PRESS, SECOND EDITION, 2010.