This paper presents a Galerkin approach for solving a regularized version of the Cauchy singular, linear integro-differential equation
\[
d
Θ
d
x
(
x
)
−
f
(
x
)
=
λ
∫
y
=
0
1
Θ
(
y
)
x
−
y
d
y
,
x
∈
(
0
,
1
)
\frac {{d\Theta }}{{dx}}\left ( x \right ) - f\left ( x \right ) = \lambda \smallint _{y = 0}^1\frac {{\Theta \left ( y \right )}}{{x - y}}dy, \qquad x \in \left ( 0, 1 \right )
\]
, subject to
Θ
(
0
)
=
Θ
(
1
)
=
0
\Theta \left ( 0 \right ) = \Theta \left ( 1 \right ) = 0
. This equation has appeared in both combined infrared gaseous radiation and molecular conduction, and elastic contact studies. A regularized formulation is produced which suggests the use of an expansion technique where the orthogonal basis functions are chosen as the Chebychev polynomials of the first kind. Accurate results, requiring a minimal computational cost, are formally documented and compared to a purely numerical solution.