514 DIFFERENTIAL EQUATIONS

The partial differential equation

∂^2 φ

∂x^2

+

∂^2 φ

∂y^2

+

∂^2 φ

∂z^2

=0 is calledLaplace’s equation.

Ifφ(x,y,z)=

1

√

x^2 +y^2 +z^2

=(x^2 +y^2 +z^2 )−

1

2

then differentiating partially with respect toxgives:

∂φ

∂x

=−

1

2

(x^2 +y^2 +z^2 )−

3

(^2) (2x)
=−x(x^2 +y^2 +z^2 )−
3
2
and
∂^2 φ
∂x^2
=(−x)
[
−
3
2
(x^2 +y^2 +z^2 )−
5
(^2) (2x)
]
+(x^2 +y^2 +z^2 )−
3
(^2) (−1)
by the product rule

3 x^2
(x^2 +y^2 +z^2 )
5
2
−
1
(x^2 +y^2 +z^2 )
3
2

(3x^2 )−(x^2 +y^2 +z^2 )
(x^2 +y^2 +z^2 )
5
2
Similarly, it may be shown that
∂^2 φ
∂y^2

(3y^2 )−(x^2 +y^2 +z^2 )
(x^2 +y^2 +z^2 )
5
2
and
∂^2 φ
∂z^2

(3z^2 )−(x^2 +y^2 +z^2 )
(x^2 +y^2 +z^2 )
5
2
Thus,
∂^2 φ
∂x^2

• ∂^2 φ
  ∂y^2


• 
  

  ∂^2 φ
  ∂z^2

  
  

  (3x^2 )−(x^2 +y^2 +z^2 )
  (x^2 +y^2 +z^2 )
  5
  2

  
  


• 
  

  (3y^2 )−(x^2 +y^2 +z^2 )
  (x^2 +y^2 +z^2 )
  5
  2

  
  


• 
  

  (3z^2 )−(x^2 +y^2 +z^2 )
  (x^2 +y^2 +z^2 )
  5
  2

  
  

  ⎛
  ⎜
  ⎜
  ⎝
  3 x^2 −(x^2 +y^2 +z^2 )

  
  


• 3 y^2 −(x^2 +y^2 +z^2 )


• 3 z^2 −(x^2 +y^2 +z^2 )
  ⎞
  ⎟
  ⎟
  ⎠
  (x^2 +y^2 +z^2 )
  5
  2
  = 0
  Thus,
  1
  √
  x^2 +y^2 +z^2
  satisfies the Laplace equation
  ∂^2 φ
  ∂x^2


• 
  

  ∂^2 φ
  ∂y^2

  
  


• 
  

  ∂^2 φ
  ∂z^2
  = 0
  Now try the following exercise.
  Exercise 200 Further problems on the solu-
  tion of partial differential equations by direct
  partial integration

  
  

1. Determine the general solution of
   ∂u
   ∂y

= 4 ty [u= 2 ty^2 +f(t)]

2. Solve

∂u

∂t

= 2 tcosθgiven thatu= 2 twhen

θ= 0. [u=t^2 (cosθ− 1 )+ 2 t]

3. Verify thatu(θ,t)=θ^2 +θtis a solution of
   ∂u
   ∂θ

− 2

∂u

∂t

=t.

4. Verify thatu=e−ycosxis a solution of
   ∂^2 u
   ∂x^2

+

∂^2 u

∂y^2

=0.

5. Solve

∂^2 u

∂x∂y

=8eysin 2xgiven that aty=0,

∂u

∂x

=sinx, and atx=

π

2

,u= 2 y^2.

[u=−4eycos 2x−cosx+4 cos 2x

+ 2 y^2 −4ey+ 4

]

6. Solve

∂^2 u

∂x^2

=y(4x^2 −1) given that atx=0,

u=sinyand

∂u

∂x

=cos 2y.

[

u=y

(

x^4

3

−

x^2

2

)

+xcos 2y+siny

]