Hmm, when I do the same insertion, I get:

$\displaystyle {C_0 \over k} L^k = C_L(L-x)^k$

$\displaystyle C_L = {{C_0 \over k} L^k \over (L-x)^k}$

$\displaystyle C_L = {C_0 \over k} \left({L \over L-x}\right)^k$

Substituting k = 0.15 gives:

$\displaystyle C_L = {C_0 \over 0.15} \left({L \over L-x}\right)^{0.15}$

Yes, there is a difference, which suggests the original formulation is incorrect.

Apparently, there should be a (1-k) somewhere instead of a k.

I haven't tried to understand your process yet, but is it possible it should have been:

$\displaystyle (1-k)C_L dx=(L-x) dC_L$