Question

Find the partial differential equation of all spheres of given radius

Answer 1

Step-by-step explanation:

Formation of Partial Differential Equations

Partial differential equations can be obtained by the elimination of arbitrary constants or by the elimination of arbitrary functions.

By the elimination of arbitrary constants

Let us consider the function

f( x, y, z, a, b ) = 0 ------------- (1)

where a & b are arbitrary constants

Differentiating equation (1) partially w.r.t x & y, we get

Eliminating a and b from equations (1), (2) and (3), we get a partial differential equation of the first order of the form f (x,y,z, p, q) = 0

Example 1

Eliminate the arbitrary constants a & b from z = ax + by + ab

Consider z = ax + by + ab ____________ (1)

Differentiating (1) partially w.r.t x & y, we get

Using (2) & (3) in (1), we get

z = px +qy+ pq

which is the required partial differential equation.

Example 2

Form the partial differential equation by eliminating the arbitrary constants a and b from

z = ( x2 +a2 ) ( y2 + b 2)

Given z = ( x2 +a2 ) ( y2 + b2) ……..(1)

Differentiating (1) partially w.r.t x & y , we get

p = 2x (y2 + b2 )

q = 2y (x + a )

Substituting the values of p and q in (1), we get

4xyz = pq

which is the required partial differential equation.

Example 3

Find the partial differential equation of the family of spheres of radius one whose centre lie in the xy - plane.

The equation of the sphere is given by

( x –a )2 + ( y- b) 2 + z2 = 1 _____________ (1)

Differentiating (1) partially w.r.t x & y , we get

2 (x-a ) + 2 zp = 0

2 ( y-b ) + 2 zq = 0

From these equations we obtain

x-a = -zp _________ (2)

y -b = -zq _________ (3)

Using (2) and (3) in (1), we get

z2p2 + z2q2 + z 2 = 1

or z2 ( p2 + q2 + 1) = 1

Example 4

Eliminate the arbitrary constants a, b & c from

and form the partial differential equation.

The given equation is

or -zp + xzr + p2x = 0

By the elimination of arbitrary functions

Let u and v be any two functions arbitrary function. This relation can be expressed as

u = f(v) ______________ (1)

Differentiating (1) partially w.r.t x & y and eliminating the arbitrary functions from these relations, we get a partial differential equation of the first order of the form

f(x, y, z, p, q ) = 0.

Example 5

Obtain the partial differential equation by eliminating „f„from z = ( x+y ) f ( x2 - y2 )

Let us now consider the equation

z = (x+y ) f(x2- y2) _____________ (1)

Differentiating (1) partially w.r.t x & y , we get

p = ( x + y ) f ' ( x2 - y2 ) . 2x + f ( x2 - y2 )

q = ( x + y ) f ' ( x2 - y2 ) . (-2y) + f ( x2 - y2 )

i.e, py - yf( x2 - y2 ) = -qx +xf ( x2 - y2 )

i.e, py +qx = ( x+y ) f ( x2 - y2 )

Therefore, we have by(1), py +qx = z

Example 6

Form the partial differential equation by eliminating the arbitrary function f

from

z = ey f (x + y)

Consider z = ey f ( x +y ) ___________ ( 1)

Differentiating (1) partially w .r. t x & y, we get

p = ey f ' (x + y)

q = ey f '(x + y) + f(x + y). ey

Hence, we have

q = p + z

Example 7

Answer 2

Answer:

The order of a pdf is the order of the ____ partial differential coefficient in it.