
Analytic solutions of variational problems 79
Let us now consider simply using pre-computed complete integrals. Cer-
tain complete integral solutions actually contain integrals. Consider the non-
homogeneous differential equation type with non-constant coefficients
a(x)(
∂u
∂x
)
2
+ b(x)(
∂u
∂y
)
2
= f (x)+g(y).
Such problems have a complete integral solution of
u(x, y)=
x
f(t)+a
1
a(t)
dt +
y
g(t) − a
1
b(t)
dt + a
2
.
For example, the equation
(
∂u
∂x
)
2
+(
∂u
∂y
)
2
= x + y
has a complete integral solution of the form
u(x, y)=
x
√
t + a
1
dt +
y
√
t − a
1
dt + a
2
.
There are also rather specific, but practical problems where the partial deriva-
tives occur in an exponential expression. The ...