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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 of rate 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

$$\frac{dx}{dt} = k(a-x)(b-x)$$

I need to solve it with given initial conditions.

I assume $b$ is different from $a$.

Therefore, $x = \frac{ab[exp (b - a)kt - 1]}{[b exp (b - a)kt - a]}$

And if $b = a$, $x = \frac{[(a^2)kt]}{(akt + 1)}$

Have no idea how to proceed further.

Any help would be appreciated.

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    What "proceed further" would you need to do? It seems to me that you are more or less done, except that your solution doesn't involve $x_0$, which it should. If you repeat essentially the same calculation, using $\int_{x_0}^x \frac{1}{k} \frac{1}{(a-x)(b-x)} dx = \int_{t_0}^t ds = t-t_0$, then you'll have your initial condition in the solution. If you don't want to do symbolic definite integration then you'll need to use an arbitrary integration constant and then solve for it (I don't recommend this). – 2017-02-23
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    By the way, note that $t_0$ is essentially yours to choose (because the equation is autonomous) while $x_0$ is given in the paragraph. – 2017-02-23

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