
1.3. Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= h(x, y)w 93
◮ Coefficients of equations contain other p ower-law functions.
16. a
∂w
∂x
+ b
∂w
∂y
= (cx
3
+ dy
3
)w.
General solution: w = exp
bcx
4
+ ady
4
4ab
Φ(bx − ay).
17. x
∂w
∂x
+ y
∂w
∂y
= a
p
x
2
+ y
2
w.
General solution: w = exp
a
p
x
2
+ y
2
Φ
y
x
.
18. x
2
∂w
∂x
+ xy
∂w
∂y
= y
2
(ax + by)w.
General solution: w = exp
(ax + by)y
2
2x
Φ
y
x
.
19. x
2
y
∂w
∂x
+ axy
2
∂w
∂y
= (bxy + cx + dy + k)w.
General solution:
w =
x
b
exp
−
k
(a + 1)xy
−
d
x
−
c
ay
Φ
x
−a
y
if a 6= −1,
exp
k
xy
+ b
ln |x| +
c
y
−
d
y
Φ(xy) if a = −1.
20. axy
2
∂w
∂x
+ bx
2
y
∂w
∂y
= (any
2
+ bmx
2
)w.
General solution: w = x
n
y
m
Φ(ay
2
− bx
2
).
21. x
3
∂w
∂x
+ ay
3
∂w
∂y
= x
2
(bx + cy)w.
General solution:
w = exp
c