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 difficult for metime-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.