Two substances, A and B, are being converted into a single compound C. In the laboratory it has been shown that for these substances, the following law of conversion holds: The time rate of change of the amount x of compound C is proportional to the product of the amounts of unconverted substances A and B. Assume the units of measure so chosen that one unit of compound C is formed from the combination of one unit of A with one unit of B. If at time t = 0 there are a units of substance A, b units of substance B, and none of compound C present, show that the law of conversion may be expressed by the equation

Solve this equation with the given initial conditions.

dx dt =k(a - x)(b-x)