
1.1. Equations of the Form f(x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= 0 33
32. sinh(λy)
∂w
∂x
+ a cosh(βx)
∂w
∂y
= 0.
Principal integral: Ξ = β cosh(λy) − aλ sinh(βx).
33.
ax
n
cosh
m
(λy) + bx
∂w
∂x
+ sinh
k
(βy)
∂w
∂y
= 0.
Principal integral:
Ξ = x
1−n
E + (n − 1)a
Z
cosh
m
(λy)E dy
sinh
k
(βy)
, E = exp
b(n − 1)
Z
dy
sinh
k
(βy)
.
1.1.4 Equations Containing Logarithmic Functions
◮ Coefficients of equations contain logarithmic functions.
1.
∂w
∂x
+
a ln
k
(λx) + b
∂w
∂y
= 0.
Principal integral: Ξ = y − bx − a
Z
ln
k
(λx) dx.
2.
∂w
∂x
+
a ln
k
(λy) + b
∂w
∂y
= 0.
Principal integral: Ξ = x −
Z
dy
a ln
k
(λy) + b
.
3.
∂w
∂x
+ a ln
k
(λx) ln
n
(µy)
∂w
∂y
= 0.
Principal integral: Ξ = a
Z
ln
k
(λx) dx −
Z
dy
ln
n
(µy)
.
4.
∂w
∂x
+ a ln
k
(x + λy)
∂w