Final answer:
To solve the differential equation (dx)/(dt) = x(x-1), we separate the variables, integrate, and consider two cases. The solutions are x = 0, x = 1, x = -1, and x = 2. The correct answer is option (b) x = -1 and x = 2.
Step-by-step explanation:
To solve the differential equation (dx)/(dt) = x(x-1), we can separate the variables and integrate. Start by moving all the terms involving x to one side: (1/x)(dx) = (x-1)dt. We can now integrate both sides. The integral of (1/x)dx is ln|x| + C, where C is the constant of integration.
On the right side, we can integrate (x-1)dt to get (1/2)x^2 - x + C2, where C2 is another constant of integration. Therefore, we have ln|x| = (1/2)x^2 - x + C3, where C3 = C - C2 is the overall constant of integration. Taking the exponential of both sides, we get |x| = e^((1/2)x^2 - x + C3), which simplifies to |x| = Ce^((1/2)x^2 - x), where C = e^(C3) is the final constant.
Since we can't have negative values inside the absolute value, we consider the two cases: x = Ce^((1/2)x^2 - x) and -x = Ce^((1/2)x^2 - x). Case 1: x = Ce^((1/2)x^2 - x). This equation tells us that x = 0 and x = 1 are possible solutions, since e^((1/2)x^2 - x) is always positive. Case 2: -x = Ce^((1/2)x^2 - x). Dividing both sides by -1, we get x = -Ce^((1/2)x^2 - x). This equation is satisfied when x = -1 and x = 2, as C can be either positive or negative. Therefore, the solutions to the differential equation are x = 0, x = 1, x = -1, and x = 2. The correct answer is option (b) x = -1 and x = 2.