Discussion
Consider the differential equation
\[ \frac{dy}{dx} - y + y^{2} = 0. \]
Then \(\displaystyle\lim_{x \to \infty} y(x)\) equals.
Answer: B
The limit is 1.
Rewrite the equation as \(y' = y - y^{2} = y(1 - y)\), the logistic equation. Its equilibria are 0 and 1. For \(0 \lt y \lt 1\), \(y' \gt 0\), and for \(y \gt 1\), \(y' \lt 0\), so every solution that starts with \(y(0) \gt 0\) moves towards 1:
\[ y(x) = \frac{1}{1 + Ce^{-x}} \to 1 \]
The key takes this stable equilibrium as the answer. The value 0 is the unstable equilibrium, reached only by the zero solution, and a solution starting below 0 runs off to \(-\infty\) rather than to \(-1\); \(+\infty\) is never approached.
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