the mutual inductance of the infinite straight wire and the coil  is M, while the self inductance of the coil is L. The current in infinite wire is varying according to the relation I₁ = t, where  is a constant and t is time. then what is the time dependence of the current in the coil ?

 

but i am not sure about this but

when current start increasing in the straight wire as I1=at there is an induced current set up in the coil due to mutual induction

I=-(m/R)d(I1)/dt

which seems to be independent of time but now the sudden increase in current there is self induction and thus the current decreases and this induces a current on the straight conductor keeps changing try solving this i hope it explained the above problem!!!