082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

834 Chapter 25 Functions of several variables
We use the product ruletofind ∂2 φ
∂x . 2
∂ 2 φ
∂x = 2 −1(x2 +y 2 +z 2 ) −3/2 −x ( )
− 3 2 (x 2 +y 2 +z 2 ) −5/2 2x
= (x 2 +y 2 +z 2 ) −5/2 [−(x 2 +y 2 +z 2 ) +3x 2 ]
By a similaranalysis we have
∂ 2 φ
∂y 2 = (x2 +y 2 +z 2 ) −5/2 [−(x 2 +y 2 +z 2 ) +3y 2 ]
∂ 2 φ
∂z = 2 (x2 +y 2 +z 2 ) −5/2 [−(x 2 +y 2 +z 2 ) +3z 2 ]
So
∂ 2 φ
∂x + ∂2 φ
2 ∂y + ∂2 φ
2 ∂z = 2 (x2 +y 2 +z 2 ) −5/2 [−3(x 2 +y 2 +z 2 ) +3x 2 +3y 2 +3z 2 ]
= 0
1
Hence φ = √ isasolution of the three-dimensional Laplace’s equation.
x2 +y 2 +z2 Thetransmissionequationisanotherimportantp.d.e.Thepotential,u,inatransmission
cablewith leakagesatisfies a p.d.e. ofthe form
∂ 2 u u
∂x 2 =A∂2 ∂t +B∂u 2 ∂t +Cu
whereA,BandC areconstants relating tothe physical properties of the cable.
Theanalyticalandnumericalsolutionofp.d.e.sisanimportanttopicinengineering.
Coverage isbeyond the scope ofthis book.
EXERCISES25.5
1 Verify that
u(x,y) =x 2 +xy
is asolution ofthe p.d.e.
2 Verify that
∂u
∂x −2∂u ∂y =y
φ = sin(xy)
satisfiesthe equation
3 Verify that
∂ 2 φ
∂x 2 + ∂2 φ
∂y 2 +(x2 +y 2 )φ=0
u(x,y) =x 3 y +xy 3
isasolution ofthe equation
4 Verify that
xy ∂2 u
∂x∂y +x∂u ∂x +y∂u ∂y = 7u
u(x,y) =xy + x y
isasolution of
5 Verify that
satisfies
y ∂2 u
∂y 2 +2x ∂2 u
∂x∂y = 2x
φ(x,y)=xsiny+e x cosy
∂ 2 φ
∂x 2 + ∂2 φ
∂y 2 =−xsiny