Page 295 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes i.e. on expansion
F F 1 F F 1 F F 1 F F 1 F F 1 F F 1
x p p x x p p x x p p x = 0 ...(12)
1 1 1 1 2 2 2 2 3 3 3 3
The equation (12) is generally written as F ,F 1 0.

Similarly
, F F 0 and F ,F 0.
2 1 2
But these are linear equations having more than two independent variables. Here we have the
following rule.

Try to find two independent integrals, F = a and F = a , of the subsidiary equations
1 1 2 2
dx dp dx dp x dp
1 1 2 2 3 = 3 ...(13)
F F F F F F
p 1 x 1 p 2 x 2 p 3 x 3
If F , F satisfy the conditions
1 2
F F F F
F 1 ,F 2 1 2 1 2 0,
r 1,2,3 x r p r p r x r
and if the p’s can be found as functions of the x’s from
F = F  a = F  a = 0,
1 1 2 2
then integrate the equation formed by substituting these functions in
dz = p dx + p dx + p dx .
1 1 2 2 3 3
Examples of Jacobi Method

1. Solve
2
2p x x 3p x 2 p p 0
2 3
1 1 3
2 3
Solution:
2
Let F 2p x x 3p x 2 p p 0 ...(1)
1 1 3 2 3 2 3
The subsidiary equations are

dx 1 dp 1 dx 2 dp 2 dx 3 dp 3
F F F F F F
(2)
p 1 x 1 p 2 x 2 p 3 x 3
Now

F F F 2 F
2x x , 2p x , 3x 3 2p p , 0,
1 3
1 3
2 3
p 1 x 1 p 2 x 2
F 2 F
p 2 , 2p x 6p x
2 3
1 1
p 3 x 3
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