Question

Using the separate-and-integrate method, find the general solution to the differential equation: dx/dt = 2(1-x). Show all of your work and the steps involved in finding the general solution.

Please provide the detailed solution for this differential equation using the separate-and-integrate method.

Answer 1

To solve the differential equation dx/dt = 2(1-x), variables x and t are separated, integrated, and then x is isolated to find the general solution x = 1 - e^-(2t+C).

Step-by-step explanation:

To solve the differential equation dx/dt = 2(1-x) using the separate-and-integrate method, we take the following steps:

1. Separate the variables x and t. We rearrange the terms to get dx/(1-x) = 2 dt.
2. Integrate both sides. The integral of dx/(1-x) with respect to x is -ln|1-x|, and the integral of 2 dt with respect to t is 2t. Including the constant of integration C, we get -ln|1-x| = 2t + C.
3. Solve for x to find the general solution. Exponentiate both sides to remove the logarithm, leading to 1-x = e-(2t+C).
4. Isolate x. We then have x = 1 - e-(2t+C). This expression represents the general solution to the original differential equation.
5. To verify this is the correct solution, you can differentiate x = 1 - e-(2t+C) with respect to t and confirm that it satisfies dx/dt = 2(1-x).