Let $y$ be the solution of the differential equation $\frac{dy}{dx} = -2x(y-1)$ with $f(0) = 1$, then $\lim_{x \to \infty} f(x)$ is equal to
Step-by-Step Solution
Key Concept: Solve separable differential equations by separating variables and integrating, then apply initial conditions to find the particular solution.
Given $\frac{dy}{dx} = -2x(y-1)$, we separate variables: $\frac{dy}{y-1} = -2x\,dx$. Integrating both sides: $\ln|y-1| = -x^2 + C$, so $y = e^{-x^2+C} + 1 = Ae^{-x^2} + 1$. Using the initial condition $y(0) = 1-1+C = 0$ gives $C = 0$. Therefore $y = e^{-x^2}$, and $\lim_{z\to\infty} f(z) = \lim_{z\to\infty} e^{-z^2} = 0$.
Correct Answer: 1