Stuck.

I’m stuck on a Morin problem.

Consider a semi-infinite stick (one that extends infinitely in one direction) with a linear mass density \(\rho(x)\). The stick enjoys the property that it can be cut anywhere, have a fulcrum placed at a distance \(b\) away from the cut, and balance on the fulcrum. Find \(\rho(x)\).

Here is what I have so far.

Suppose that the end of the stick is at \(x=0\). Let a cut be made at \(x=c\). Then, the fulcrum is placed at \(x=c+b\). Balancing torques about the fulcrum: \[\int_{c}^{c+b}{(c+b-x)\rho(x)\textrm{ d}x}=\int_{c+b}^{\infty}{(x-c-b)\rho(x)\textrm{ d}x}\] This can be rearranged to: \[\int_{c}^{\infty}{(c+b-x)\rho(x)\textrm{ d}x}=0\] This is an integral equation. I have no ideas as to how to solve it. I refuse to look at the solution just yet.