ABSTRACT
This is a special case of equation 8.4.1.1 with f (w) = cwp.
2. ∂2w
∂t2 =
∂
∂x
( axn
∂w
∂x
) + ∂
∂y
( bym
∂w
∂y
) + cwp.
This is a special case of equation 8.4.1.2 with f (w) = cwp. 1◦. Suppose w(x, y, t) is a solution of the equation in question. Then the functions
w1 = C1w ( C p-1 2-n
) ,
where C1 and C2 are arbitrary constants, are also solutions of the equation. 2◦. Solution for n ≠ 2, m ≠ 2, and p ≠ 1:
w =
[ 1
2c(p – 1) (
1 + p 1 – p
+ 2
2 – n + 2
2 – m
a(2 – n)2 + y2-m
b(2 – m)2 – 1 4 (t + C)
3◦. Solution for n ≠ 2 and m ≠ 2 (generalizes the solution of Item 2◦):
w = w(r), r2 = 4k [
a(2 – n)2 + y2-m
b(2 – m)2 – 1 4 (t + C)
,
where C and k are arbitrary constants (k ≠ 0) and the function w(r) is determined by the ordinary differential equation
w′′rr + A
r w′r + ck
–1wp = 0, A = 2(4 – n – m)(2 – n)(2 – m) .
