
108 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
18. x
∂w
∂x
+ y
∂w
∂y
= ax arccot(λx + βy)w.
General solution:
w = exp
ax arccot(λx + βy) +
ax
2(λx + βy)
ln
x
2
+
x
2
(λx + βy)
2
Φ
y
x
.
19. a
∂w
∂x
+ b arccot
n
(λx)
∂w
∂y
=
c arccot
m
(µx) + s arccot
k
(βy)
w.
This is a special case of equation 1.3.7.33 with f(x) = a, g
1
(x) ≡0, g
0
(x) = b arccot
n
(λx),
and h(x, y) = c arccot
m
(µx) + s arccot
k
(βy).
20. a
∂w
∂x
+ b arccot
n
(λy)
∂w
∂y
=
c arccot
m
(µx) + s arccot
k
(βy)
w.
This is a special case of equation 1.3.7.19 with f (x) = a, g(y) = b arccot
n
(λy), h
1
(x) =
c arccot
m
(µx), and h
2
(y) = s arccot
k
(βy).
1.3.7 Equations Containing Arbitrary Functions
◮ Coefficients of equations contain ...