
15.2. Integral Transform Method 1173
into account the relation F{∂
xxxx
w} = u
4
ˇw (see Property 6 with n = 4 in Table 15.4), we arrive at
the following pro blem for a linear second-order ordina ry differential equation in t with parameter u:
ˇw
′′
tt
+ a
2
u
4
ˇw = 0,
ˇw = F (u) at t = 0, ˇw
′
t
= 0 at t = 0,
(15.2.2.11)
where F (u) = F {f (x)}. The solution of problem (15.2.2.11) has the form
ˇw = F (u) c os(au
2
t). (15.2.2.12)
We apply the inverse Fourier transform (15.2.2.4) to (15.2.2.12) and obtain, after easy tra nsforma-
tions,
w =
1
2π
Z
∞
−∞
F (u) cos(au
2
t)e
iux
du
=
1
2π
Z
∞
−∞
Z
∞
−∞
f(ξ)e
−iuξ
dξ
cos(au
2
t)e
iux
du
=
1
2π
Z
∞
−∞
f(ξ)
Z
∞
−∞
cos(au
2
t)e
iu(x−ξ)
du
dξ
=
1
π
Z
∞
−∞
f(ξ)
Z
∞
0
cos(