Page 243 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
u v v u u v v u u v v u
or p q 0
z y z y x z x z x y x y
u
v
v
u
( , ) ( , ) ( , )
u
v
p q ...(13)
z
y
y
( , ) ( , ) ( , )
x
z
x
which is partial differential equation of the type (11). It should be noted that equation (13) is a
linear partial differential equation i.e. the powers of p and q are both unity. Whereas the partial
differentiation equation (11) need not be linear. To see that consider the equation
2
2
(x a) + (y b) + z 2 = 1 ...(14)
Differentiating (14) with respect to x and y separately, we have
2(x a) + 2zp = 0, 2(y b) + 2zq = 0
Substituting the values of (x a) and (y b) in equation (14) we have
2 2
2
2
2
2
2 2
z p + z q + z = 1 or z (p + q + 1) = 1. ...(15)
So powers of p and q are not one.
Example 2: Eliminate the constants a and b from
2
2z = (ax + y) + b ...(1)
Solution: Differentiate with respect to x we have
z
2 = 2p 2 (ax y )
a
x
Differentiating (1) with respect to y we have
z
2 = 2q 2(ax y )
y
or p = a(ax + y) ...(2)
q = (ax + y) ...(3)
px + qy = ax(ax + y) + y(ax + y)

2
= (ax + y) = q 2
or px + qy = q 2
is the answer.

Example 3: Eliminate the arbitrary function f from the equation

xy
z = f ...(4)
z
Differentiating with respect to x and y respectively we have

z y xy
p = f p ...(15)
x z z 2

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