1.4 Coordinate Transformations Between Rectangular …
43
2 u =
∂
4 u
∂ x 4 + 2
∂
4 u
∂ x 2 ∂ y 2 +
∂
4 u
∂ y 4 =
d
4 u
dr 4 +
2
r
d
3 u
dr 3 −
1
r 2
d
2 u
dr 2 +
1
r 3
du
dr
(1.4.16)
Let
2 u = f (r ), then Eq. (1.4.16) can be written as
2 u =
d
4 u
dr 4 +
2
r
d
3 u
dr 3 −
1
r 2
d
2 u
dr 2 +
1
r 3
du
dr
= f (r )
(1.4.17)
Making transformation r = e
t , namely t = ln r , derive to r , we obtain
dt
dr
=
1
r
,
thus
du
dr
=
du
dt
dt
dr
=
1
r
du
dt
(1.4.18)
d
2 u
dr 2 =
1
r 2
d
2 u
dt 2 −
du
dt
(1.4.19)
d
3 u
dr 3 =
1
r 3
d
3 u
dt 3 − 3
d
2 u
dt 2 + 2
du
dt
(1.4.20)
d
4 u
dr 4 =
1
r 4
d
4 u
dt 4 − 6
d
3 u
dt 3 + 11
d
2 u
dt 2 − 6
du
dt
(1.4.21)
Substituting Eqs. (1.4.18)–(1.4.21) into Eq. (1.4.17), we obtain
1
r 4
d
4 u
dt 4 − 4
d
3 u
dt 3 + 4
d
2 u
dt 2
= f (r )
(1.4.22)
or
d
4 u
dt 4 − 4
d
3 u
dt 3 + 4
d
2 u
dt 2 = e
4t f (e
t
)
(1.4.23)
The homogeneous equation corresponding to Eq. (1.4.23) is
d
4 u
dt 4 − 4
d
3 u
dt 3 + 4
d
2 u
dt 2 = 0
(1.4.24)
Equation (1.4.24) is fourth order differential equation with constant coefficients,
its characteristic equation is
k
4
− 4k
3
+ 4k
2
= 0
(1.4.25)
Solve Eq. (1.4.15), we obtain two pairs of dual real roots k = 0 and k = 2. Thus,
the solution of the homogeneous equation is