Question

If the solution of \( \frac{dy}{dx} = y \log_e 0.5 = 0 \), \( y(0) = 1 \), and \( y(x) \to k \) as \( x \to \infty \), then \( k = \)

• A. \( \infty \)
• B. \( -1 \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

Exponential decay equations with negative exponents always tend to zero as \( x \to \infty \).

Answer 1

The Correct Option is D

Given the differential equation: \[ \frac{dy}{dx} = y \log_e(0.5) \] Since \( \log_e(0.5)<0 \), the equation becomes: \[ \frac{dy}{dx} = -a y \quad \text{where } a>0 \] The solution to this differential equation is: \[ y(x) = Ce^{-ax} \] Using the initial condition \( y(0) = 1 \Rightarrow C = 1 \), so: \[ y(x) = e^{-ax} \] As \( x \to \infty \), \( y(x) \to 0 \). So \( k = 0 \).