Page 346 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 20: Higher Order Equations with Constant Coefficients and Monge’s Method

Integrating, Notes
3y 3x 2p = a , 5y 3x 2q b ...(2)
The intermediate integral is

3y 3x 2p = f ( 5y 3x 2 ) ...(3)
q
From (2),
1 1
p = (3y 3x a ), q ( 5y 3x b )
2 2
Putting these values of p and q in
dz = p dx + q dy
1 1
dz = (3y 3x a )dx ( 5y 3x b )dy
2 2

or 2 dz = 3(ydx xdy ) 3x dx 5y dy a dx b dy
Integrating

3 2 5 2
2z = 3xy x y ax by c
2 2
This is the required complete integral of (1).
Self Assessment

19. Solve
2s (rt s 2 ) 1

20. Solve

3r 4s t (rt s 2 ) 1

20.10 Summary

 The partial differential equations are classified according to their structure.

 Similar method as used in ordinary differential equations is adopted for partial differential
equations with constant coefficients.
 The methods, adopted in solving various equations are given in details. It is advisable to
understand the partial differential equations and apply the appropriate methods.

20.11 Keywords

C.F. or Complimentary Function is the solution of the partial differential equations containing
a number of arbitrary constants.
P.I. or Particular Integral is the particular solution of the partial differential equation containing
any arbitrary constants.

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