
4.3. Other Equations 461
6.
∂w
∂t
= a
1
(t)
∂
2
w
∂x
2
+ a
2
(t)
∂
2
w
∂y
2
+ [b
1
(t)x + c
1
(t)]
∂w
∂x
+ [b
2
(t)y + c
2
(t)]
∂w
∂y
+ [s
1
(t)x
2
+ s
2
(t)y
2
+ p
1
(t)x + p
2
(t)y + q(t)]w.
The substitution
w(x, y, t) = exp
f
1
(t)x
2
+ f
2
(t)y
2
u(x, y, t),
where the functions f
1
= f
1
(t) and f
2
= f
2
(t) are solutions of the Riccati equations
f
′
1
= 4a
1
(t)f
2
1
+ 2b
1
(t)f
1
+ s
1
(t),
f
′
2
= 4a
2
(t)f
2
2
+ 2b
2
(t)f
2
+ s
2
(t),
leads to an equation of the form 4.3.2.5 for u = u(x, y, t).
7.
∂w
∂t
= a
1
(x)
∂
2
w
∂x
2
+ a
2
(y)
∂
2
w
∂y
2
+ b
1
(x)
∂w
∂x
+ b
2
(y)
∂w
∂y
+ [c
1
(x) + c
2
(y)]w + Φ(x, y, t).
Domain: x
1
≤ x ≤ x
2
, y
1
≤ y ≤ y
2
. Different boundary value problems:
w = f (x, y) at t = 0 (initial condition),
s
1
∂
x
w − k
1
w = g
1
(y, t) at x = x
1
(boundary ...