Question

If the curve \(y=f(x)\) passes through the point \((1,e)\) and satisfies the differential equation} \[ dy=y(2+\log_e x)\,dx,\quad x>0, \] then \(f(e)\) is equal to:

• A. \(e^{e}\)
• B. \(e^{e^2}\)
• C. \(e^{2e}\)
• D. \(e^{2e}\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The given equation is a first-order variable separable differential equation. We can separate the variables \(y\) and \(x\) to integrate both sides.
Step 2: Key Formula or Approach:
1. \(\int \frac{1}{y} dy = \int (2 + \log_e x) dx\).
2. Integration by parts: \(\int \log_e x dx = x \log_e x - x\).
Step 3: Detailed Explanation:
Separate the variables:
\[ \frac{dy}{y} = (2 + \log_e x) dx \]
Integrate both sides:
\[ \int \frac{dy}{y} = \int 2 dx + \int \log_e x dx \]
\[ \log_e y = 2x + (x \log_e x - x) + C \]
\[ \log_e y = x + x \log_e x + C \]
The curve passes through \((1, e)\). Substitute \(x=1, y=e\):
\[ \log_e e = 1 + 1 \log_e 1 + C \]
\[ 1 = 1 + 0 + C \implies C = 0 \].
The curve is \(\log_e y = x(1 + \log_e x)\).
To find \(f(e)\), substitute \(x=e\):
\[ \log_e y = e(1 + \log_e e) \]
\[ \log_e y = e(1 + 1) = 2e \]
\[ y = e^{2e} \].
Step 4: Final Answer:
The value of \(f(e)\) is \(e^{2e}\).