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: Rearrange the equation.
From $\dfrac{dy}{dx}-y\log_e 0.5=0$, we get $\dfrac{dy}{dx}=y\log_e 0.5$, a separable equation.
Step 2: Separate variables.
$\dfrac{dy}{y}=\log_e 0.5\,dx$.
Step 3: Integrate both sides.
$\log_e y=(\log_e 0.5)\,x+C$.
Step 4: Solve for $y$.
Exponentiating, $y=A\,e^{(\log_e 0.5)x}=A(0.5)^x$, where $A=e^{C}$.
Step 5: Apply $y(0)=1$.
At $x=0$, $1=A(0.5)^0=A$, so $A=1$ and $y=(0.5)^x$.
Step 6: Take the limit as $x\to\infty$.
Since $0.5<1$, $(0.5)^x\to0$, so $k=0$.
\[ \boxed{0} \]