What is the solution to the equation y' = y(1 - y)?

What is the solution to the equation y' = y(1 - y)?

Step 1: Recognize that the equation y' = y(1 - y) is a separable differential equation.
Step 2: Rewrite the equation as dy/dx = y(1 - y).
Step 3: Separate the variables by dividing both sides by y(1 - y) and multiplying both sides by dx: (1 / (y(1 - y))) dy = dx.
Step 4: Integrate both sides. The left side requires partial fraction decomposition: 1/(y(1 - y)) = 1/y + 1/(1 - y).
Step 5: Integrate the left side: ∫(1/y) dy + ∫(1/(1 - y)) dy = ln|y| - ln|1 - y| + C1.
Step 6: The right side integrates to x + C2.
Step 7: Combine the results: ln|y| - ln|1 - y| = x + C, where C = C2 - C1.
Step 8: Exponentiate both sides to eliminate the natural logarithm: |y| / |1 - y| = e^(x + C).
Step 9: Let C' = e^C, then |y| / |1 - y| = C' e^x.
Step 10: Solve for y: y = C' e^x / (1 + C' e^x).
Step 11: Replace C' with C for simplicity: y = C / (C + e^x).

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