In the wikipedia page on Elliptic Integrals, under the section Complete Elliptic Integrals of the 1st Kind, sub-section "Differential Equation", is the following:-
===Differential equation===
The differential equation for the elliptic integral of the first kind is $$\frac{d}{dk}\left(k\left(1-k^2\right)\frac{dK(k)}{dk}\right)=k K(k)$$
A second solution to this equation is $$K\left(\sqrt{1-k^2}\right)$$. This solution satisfies the relation $$\frac{d}{dk}K(k)=\frac{E(k)}{k\left(1-k^2\right)}-\frac{K(k)}{k}$$.
I interpreted " a second solution to this equation is" to imply (by association of the form S1:$A=B$, S2:$A=C$ => S3:$A=C$) that:-
$$kK(k) = K(\sqrt{1-k^2})$$
However if I ask Wolfram Alpha to solve:-
$$kK(k) - K(\sqrt{1-k^2}) = ?$$
it gives an answer which is not zero.
(I get a similar problem when looking at the differential of the Complete Elliptic Integral of the 2nd kind, but I will put that aside for now).
Where am I going wrong here?