Mathematical Physics Vol 1

7.2 Formation of partial differential equations

337

Starting from Monge notation, we obtain

∂ z ∂ x

∂ z ∂ y

= 2 a 2 x + 2 aby + c , q =

= 2 b 2 y + 2 abx + d ,

p =

∂ 2 z ∂ x 2

∂ 2 z ∂ x ∂ y

∂ ∂ x

∂ ∂ x

∂ ∂ y

( p )= 2 a 2 , s =

r =

( q )=

( p )= 2 ab ,

=

=

∂ 2 z ∂ y 2

∂ ∂ y

( q )= 2 b 2 .

t =

=

From the last relations we obtain the required equation rt − s 2 = 0 .

Example 235 Determine the differential equation of the vibrating string (Fig. 5.1).

Figure 7.1: The vibrating string.

Solution Assume the following - the mass of the wire is constant per unit length (homogeneous wire), - the weight of the wire is neglected (the force of gravity is negligible in relation to the forces that occur in the wire), - the horizontal displacement of the points of the wire u is small in comparison with its length, so it can be considered that the points have only a vertical displacement. The deflections and inclinations at each point are also small. We start from Newton’s law m a = ∑ i S i , where m ismass, a acceleration, and S i forces acting on the observed point. The projections of this vector equation (in this example there are two forces, thus