Mathematical Physics Vol 1

7.2 Formation of partial differential equations

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Solution The equation of this family of spheres is

( x − a ) 2 +( y − a ) 2 +( z − b ) 2 = R 2 .

(7.10)

Considering z to be a function of x and y , and differentiating the relation (7.10) with respect to x and y , respectively, yields

∂ z ∂ x

∂ z ∂ y

2 ( x − a )+ 2 ( z − b )

= 0 , 2 ( y − a )+ 2 ( z − b )

= 0 ,

or

( x − a )+( z − b ) p = 0 , ( y − a )+( z − b ) q = 0 . Let us introduce the notation z − b = − m . It follows that x − a = pm , and y − a = qm . Substituting into the equation (7.10) yields m 2 ( p 2 + q 2 + 1 )= R 2 . (7.11) Given that ( x − y )=( p − q ) m , it follows that m = . Further, substituting m

x − y p − q

obtained it this way into (7.11), we have x − y p − q 2 that is, the required partial differential equation

( p 2 + q 2 + 1 )= R 2 ,

( x − y ) 2 ( p 2 + q 2 + 1 )= R 2 ( p − q ) 2 .

Example 237 Determine the equation of the oscillating membrane.

R Before we start solving this task, let us explain the terms that appear in it:

• A membrane is a material surface, i.e. a geometric surface to which a continuously distributed mass is assigned. • Density is the mass per unit area of the membrane. If the density is constant, then the membrane is called a homogeneous membrane. • Stress is the force per unit length of the membrane. • Slope is the angle that a tangent direction, at a point of the membrane, forms with its projection on the xy –plane. Solution Let us first state the physical conditions (constraints) under which the required equation is derived: 1 ◦ an elastic membrane is observed, whose density is constant.