Page 253 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes then (u, v) = 0 is a general integral.
Hence we have the following rule:
To obtain an integral of the linear equation of the form (1), find two independent integrals of
equation (3). Let they be denoted by u = a and v = b, then (u, v) = 0, where is an arbitrary
function, is an integral of the partial differential equation. Equations (3) are called subsidiary
equations.
The solution may also be written in the form
u = f ( ) ... (4)
where f denotes an arbitrary function of v.

This is known as Lagrange’s solution of the linear equation.
The method given above can be extended to the general equation of the form

z z
P P P z
1 2 n = R ... (5)
x x x
1 2 n
where P , P , ... P , R are functions of (x , x , ... x , z). To solve equation (5) we write the subsidiary
1 2 n 1 2 n
equations
dx 1 dx 2 dx n
P 1 P 2 = P n ... (6)

and find n independent integrals of this system of these subsidiary equations, in the form
u = c , u = c , u = c , ... u = c ... (7)
1 1 2 2 3 3 n n
then the integral of the given equation (5) is
(u , u , ... u ) = 0 ... (8)
1 2 n
17.3 Illustrative Examples

Example 1: Solve
(mz ny) p + (nx lz) q = ly mx ... (1)
Solution:

Here P = mz ny
Q = nx lz
R = ly mx
The subsidiary equations are

dx dy dz
... (2)
mz ny nx lz ly mx
 dx mdy ndz
or  (mz ny ) m (nx  ) z ( n y mx )

 dx mdy ndz  dx mdy ndz
or  mz  ny mnx m z n y nmx O



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