
118 FIRST-ORDER EQUATIONS WITH TWO INDEPENDENT VARIABLES
◮ Coefficients of equations contain arbitrary powers of x and y.
24. a
∂w
∂x
+ b
∂w
∂y
= cw + kx
n
y
m
.
Two forms of the representation of the general solution:
w = exp
c
a
x
Φ(bx −ay) +
k
a
m+1
Z
x
n
(bx − u)
m
exp
−
c
a
x
dx
,
w = exp
c
b
y
Φ(bx − ay) +
k
b
n+1
Z
y
m
(ay + u)
n
exp
−
c
b
y
dy
,
where u = bx − ay. In the integration, u is considered a parameter.
25. a
∂w
∂x
+ y
∂w
∂y
= bw + cx
n
y
m
.
General solution:
w = y
b
Φ(y
a
e
−x
) + c
Z
y
m−b−1
(a ln y − ln u)
n
dy
, where u = y
a
e
−x
.
In the integration, u is considered a parameter.
26. x
∂w
∂x
+ y
∂w
∂y
= axw + bx
n
y
m
.
General solution: w = e
ax
Φ
y
x
+ bx
−m
y
m
Z
x
m+n−1
e
−ax
dx
.
27. x
∂w
∂x
+ y
∂w
∂y
= a
p
x
2
+ y
2