Page 241 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
f f f
: :
x y z
f f z
Also p
x z x

f f z
and q
y z y
The symbols p, q are to be understood as here defined.
Theorem 2: The envelope of the system of surfaces

f(x, y, z, a, b) = 0,
where a, b are variable parameters, is found by eliminating a and b by using the given relation
f f
and 0, 0.
a b

Example 1: Let us consider the equation
2
2
x + y + (z c) 2 = a 2 ...(1)
which contains two constants a and c. This equation represents the set of all spheres whose
centers lie along the z-axis. If we differentiate the equation (1) with respect to x, we obtain the
relation
z
2x 2(z c ) = 0 ...(2)
x
And if we differentiate the equation (1) with respect to y. We obtain the relation

z
2y 2(z c ) = 0 ...(3)
y
Eliminating (c) from equations (2) and (3) we have

z z
2x 2y = 0
y x
or xq yp = 0 ...(4)

z z
where p and q . The equation (4) is a first order partial differential equation and is
x y
linear.
We can show that there are other geometrical entities other than the set of all spheres with
centers along the z-axis which can be described by the equation (4).

Let us consider the equation
2
2
2
x + y 2 = (z c) tan ...(5)
in which the constants c and are arbitrary. Differentiating (5) with respect to x and y, we get the
relations
2
2
p(z c)tan = x, q(z c)tan = y ...(6)

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