
88 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
20. f (x)
∂w
∂x
+ g(y)
∂w
∂y
= h
1
(x) + h
2
(y).
General solution:
w =
Z
h
1
(x)
f(x)
dx +
Z
h
2
(y)
g(y)
dy + Φ
Z
dx
f(x)
−
Z
dy
g(y)
.
21. f
1
(x)
∂w
∂x
+
f
2
(x)y + f
3
(x)y
k
∂w
∂y
= g(x)h(y).
The transformation ξ =
Z
f
2
(x)
f
1
(x)
dx, η = y
1−k
leads to an equation of the form 1.2.7.19:
∂w
∂ξ
+
(1 − k)η + F (ξ)
∂w
∂η
= G(ξ)H(η),
where F (ξ) = (1 − k)
f
3
(x)
f
2
(x)
, G(ξ) =
g(x)
f
2
(x)
, and H(η) = h(y).
22. f
1
(x)g
1
(y)
∂w
∂x
+ f
2
(x)g
2
(y)
∂w
∂y
= h
1
(x)h
2
(y).
The transformation ξ =
Z
f
2
(x)
f
1
(x)
dx, η =
Z
g
1
(y)
g
2
(y)
dy leads to an equation of the form
1.2.7.18:
∂w
∂ξ
+
∂w
∂η
= F (ξ)G(η), where F (ξ) =
h
1
(x)
f
2
(x)
, G(η) =
h
2
(y)
g
1
(y)
.
23. f
1
(x)g
1
(y)
∂w
∂x
+ f
2
(x)g
2
(y)
∂w
∂y
= h
1
(