Solve the equation y' = y(1 - y).

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Question: Solve the equation y\' = y(1 - y).

Options:

Correct Answer: y = 1/(C - x)

Solution:

Separating variables and integrating gives y = 1/(C - x).

Solve the equation y' = y(1 - y).

Solve the equation y' = y(1 - y).

Step 1: Start with the equation y' = y(1 - y).
Step 2: Rewrite y' as dy/dx, so the equation becomes 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|.
Step 6: The right side integrates to x + C, where C is the constant of integration.
Step 7: Combine the results: ln|y| - ln|1 - y| = x + C.
Step 8: Use properties of logarithms to combine: ln(|y| / |1 - y|) = x + C.
Step 9: Exponentiate both sides to eliminate the logarithm: |y| / |1 - y| = e^(x + C).
Step 10: Let K = e^C, so |y| / |1 - y| = K * e^x.
Step 11: Solve for y: y = K * e^x / (1 + K * e^x).
Step 12: Rewrite K as 1/C for simplicity, giving y = 1 / (C - e^x).

Separation of Variables – This technique involves rearranging a differential equation to isolate the variables on different sides for integration.
Integration of Functions – The process of finding the integral of a function, which is essential for solving differential equations.
Equilibrium Solutions – Identifying constant solutions where y' = 0, which can provide insights into the behavior of the solution.