Question

If the solution of \[ \frac{dy}{dx}-y\log_e0.5=0,\quad y(0)=1, \] and \(y(x)\to k\), as \(x\to\infty\) then \(k=\)

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

Hint:

For differential equations of the form \[ \frac{dy}{dx}=ky, \] the solution is \[ y=Ae^{kx}. \] If \(k\lt 0\), then the solution approaches \(0\) as \(x\to\infty\).

Answer 1

The Correct Option is D

Step 1: Write the given differential equation.
The given differential equation is \[ \frac{dy}{dx}-y\log_e0.5=0 \] Rearranging, \[ \frac{dy}{dx}=y\log_e0.5 \]

Step 2: Separate the variables.
\[ \frac{dy}{y}=\log_e0.5\,dx \] Integrating both sides, \[ \int \frac{1}{y}\,dy = \int \log_e0.5\,dx \] \[ \log_e y=x\log_e0.5+C \]

Step 3: Simplify the solution.
Exponentiating both sides, \[ y=e^{x\log_e0.5+C} \] \[ y=e^C\cdot e^{x\log_e0.5} \] Let \[ e^C=A \] Then, \[ y=A(0.5)^x \]

Step 4: Use the initial condition.
Given, \[ y(0)=1 \] Substituting \(x=0\), \[ 1=A(0.5)^0 \] \[ 1=A \] Thus, \[ y=(0.5)^x \]

Step 5: Find the limiting value.
As \(x\to\infty\), \[ (0.5)^x\to0 \] Therefore, \[ y(x)\to0 \] Hence, \[ k=0 \]

Step 6: Final Answer.
Therefore, \[ \boxed{0} \]