Differential Equation: Particular Integral

"Let the particular integral s* be: s*=C\lambda + D"

Wait...are you sure the \lambda s are the same?

EDIT: don't assume I know anything about diffeqs.

This is the differential equation:
$\frac{d^2s}{d\lambda} + \frac{\lambda}{mL}s = g + \frac{\lambda}{m}$

Let the particular integral, s* be:
$s^* = C\lambda + D$
$\frac{ds^*}{d\lambda}$$= C$
$\frac{d^2s^*}{d\lambda}$$= 0$

Hence:
$\frac{\lambda}{mL}(C\lambda + D) = g + \frac{\lambda}{m}$

So:
$C\lambda + D = \frac{mgL}{\lambda} + L$

Comparing coeffficients:
$D = L$

But, is:
$C\lambda = \frac{mgL}{\lambda}?$

I know that they are the only variables remaining, but $C\lambda$'s $\lambda$ is raised to the power of 1, while $\frac{mgL}{\lambda}$'s $\lambda$ is raised to the power of -1. So, I don't see how they are equal.

Anyways, the pdf I am using for reference tells that the particular integral is:

$s^* = \frac{mgL}{\lambda} + L$

Which implies that the relationship of C with $\frac{mgL}{\lambda}$ is true.