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I'm not sure how to solve the following problem. It's quite unlike the other problems I've had to solve since I started learning about differential equations. Can anybody help?

A disease spreads at a speed that is proportional to the multiplier of healthy and sick people. Let's call the sick quotient of a group x and the quotient of healthy people 1 - x. Then we have dx / dt = kx(1 - x). If half of the group is sick and the disease spreads at a constant speed, every member of the group will be sick in 24 days time. Then we have x = 1/2 , 1 - x = 1/2 and dx / dt = (1/2)/24 = 1/48 , so 1/48 = k * 1/2 * 1/2 <=> k = 1/12 , which gives dx / dt = 1/12 x(1 - x). Calculate how big a fraction of the group is sick in 12 days time.

A disease spreads at a speed that is proportional to the multiplier of healthy and sick people. Let's call the sick quotient of a group x and the quotient of healthy people 1 - x. Then we have dx / dt = kx(1 - x). If half of the group is sick and the disease spreads at a constant speed, every member of the group will be sick in 24 days time. Then we have x = 1/2 , 1 - x = 1/2 and dx / dt = (1/2)/24 = 1/48 , so 1/48 = k * 1/2 * 1/2 <=> k = 1/12 , which gives dx / dt = 1/12 x(1 - x). Calculate how big a fraction of the group is sick in 12 days time.

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