
68 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
◮ Coefficients of equations are quadratic in x and y.
9. a
∂w
∂x
+ b
∂w
∂y
= cx
2
+ dy
2
+ kxy + n.
General solution:
w =
1
6a
2
b
b(2ac − bk)x
3
+ 2a
2
dy
3
+ 3abx(kxy + 2n)
+ Φ(bx − ay).
10. ax
∂w
∂x
+ by
∂w
∂y
= cx
2
+ dy
2
+ kxy + n.
General solution:
w =
1
2ab
(bcx
2
+ ady
2
) +
n
a
ln |x| +
k
a + b
xy + Φ
|x|
−b/a
y
if a + b 6= 0,
1
2a
(cx
2
− dy
2
) +
1
a
(kxy + n) ln |x| + Φ(xy) if a + b = 0.
11. ay
∂w
∂x
+ bx
∂w
∂y
= cxy + d.
General solution: w =
c
2a
x
2
+
d
√
ab
ln
√
ab x + ay
+ Φ(ay
2
− bx
2
).
12. ax
2
∂w
∂x
+ by
2
∂w
∂y
= cx
2
+ dy
2
+ kxy + nx + my + s.
General solution:
w =
c
a
x −
s
ax
−
dy
2
ax − by
+
kxy
ax −by
ln
ax
y
+
n
a
ln |x| +
m
b
ln |y| + Φ
ax − by
xy
.
13. x
2
∂w
∂x