Page 273 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
1
= 2 ( a x y )] 2 a (x y ) . b
4

2
2
Example 5: Solve: z (p q ) = x y.
Solution:
2 3/2
Putting Z = z
3
Z 2 3 1/2 z
z .
x 3 2 x

2 2
z Z
z = P (say)
2
x x
Similarly,
2 2
z Z
2
z = Q (say)
y y
2
2
P Q = x y.
2
Let P x = Q y = c.
P = (c + x) and Q = (c + y).
dZ = P dx + Q dy
= (c + x) dx + (c + y) dy.

(c x ) 3/2 (c y ) 3/2
Z k 1
3 3
2 2

or z 3/2 = (c + x) 3/2 + (c + y) 3/2 + k.
is the required solution.

Self Assessment

Solve the following:
2
19. q = 2yp .
2 2
2
20. x p = yq .
17.5 Summary

 Lagrange method is quite famous. It is used also in the theory of total differential equations
as well as simultaneous differential equations.

 It can be easily extended to the theory of partial differential equations involving more
than two independent variables.

266 LOVELY PROFESSIONAL UNIVERSITY