
8 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
34. (Axy + Aky + Bx
2
+ Bkx)
∂w
∂x
+
Cy
2
+ Dxy + k(D − B)y
∂w
∂y
= 0.
Principal integral:
Ξ = (x + k)E + kB
Z
E dv
v
(C − A)v + D −B
, v =
y
x + k
,
where E = exp
Z
(Av + B) dv
v
(A −C)v + B − D
.
35. (Ay
2
+ Bxy + Cx
2
+ kx)
∂w
∂x
+ (Dy
2
+ Exy + F x
2
+ ky)
∂w
∂y
= 0.
Principal integral:
Ξ = xV + k
Z
V dv
Av
3
+ (B − D)v
2
+ (C − E)v − F
, v =
y
x
,
where V = exp
Z
(Av
2
+ Bv + C) dv
Av
3
+ (B − D)v
2
+ (C − E)v − F
.
36. (Ay
2
+ Bxy + Cx
2
)
∂w
∂x
+ (Dy
2
+ Exy + F x
2
)
∂w
∂y
= 0.
Principal integral:
Ξ =
Z
(Av
2
+ Bv + C) dv
Av
3
+ (B − D)v
2
+ (C − E)v − F
+ ln |x|, v =
y
x
.
37. (Ay
2
+ 2Bxy + Dx
2
+ a)
∂w
∂x
− (By
2
+ 2Dxy − Ex
2
− b)
∂w
∂y
= 0.
Principal integral: Ξ = Ay
3
− Ex