What ideas have you had so far?
In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C: A + B -> C. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B: d[C]/dt = k[A][B]. Thus, if the initial concentrations are [A] = a moles/L and [B] = b moles/L and we write x= [C], then we have dx/dt= k(a-x)(b-x).
(a) Assuming that a does not equal b, find x as a function of t. Use the fact that the initial concentration of C is 0.
(b) Find x(t) assuming that a = b. How does this expression for x(t) simplify if it is known that [C] = (1/2)a after 20 seconds?
So, there are two checks you need to perform (which you should ALWAYS do with every single DE you ever solve):
1. Check that your solution satisfies the initial conditions.
2. Check that your solution satisfies the original DE.
Does your solution satisfy those two criteria?
I don't think so, either. Can you show all your steps from the separation of variables to your final solution, including plugging in the initial condition? Your answer is quite close to the correct answer, so I imagine it's just a small algebra error somewhere.
Close, but not quite. You have the fraction
as the integrand on the LHS of your separated DE. You would like to write this as two separate fractions. So, pretend you can and see if it works:
Did you see how that last step was done? It's just a common denominator on the RHS, and then canceling the denominators of both sides.