Chemical kinetics with the reaction of tert-butyl bromide with azide ion
You really don't have to write all that differential equations. 

Note: I'll write all my steps on a paper, since MathJax formatting is a little time-taking at the moment.

Consider the rate laws for the two reactions:


Since the $\ce{[(CH3)3C+]}$ isn't changing with time, we get


From which, you get



Plugging these in the Rate2 which I mentioned, you will get,


Which is the same as what's given in option (D).

I hope I have satisfied your queries.