Current through an ideal inductor at steady state
The voltage between the ends of an ideal inductor: U=L*(di/dt) where term di/dt means the changing rate (=amperes/second) of the current through the inductor. 

In practical inductors there's always some resistace and the equation would be U=L*(di/dt)+iR, but you declared R=0.

Your both inductors , say La=5mH and Lb=10mH have the same voltage, so

La*(d(ia)/dt)=Lb*(d(ib)/dt). That doesn't allow any other possibility that the current changing rates are inversely proportional to the inductances. Thus the cumulated currents are, too.