If \(y = y(x)\), \(x \gt 1\), \(y \gt 0\), is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = \frac{y}{x\log_e x}\) satisfying \(y(e) = 2\), then \(y(e^{3})\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 6
Separating the variables:
\( \displaystyle \frac{dy}{y} = \frac{dx}{x\log x} \)
For the right side, put \(t = \log x\), so \(dt = \frac{dx}{x}\):
\( \displaystyle \int \frac{dx}{x\log x} = \int \frac{dt}{t} = \log (\log x) \)
So
\( \log y = \log (\log x) + \log C \)
\( y = C\log x \)
Using \(y(e) = 2\): \(C \cdot 1 = 2\), so the particular solution is
\( y = 2\log x \)
At \(x = e^{3}\):
\( y(e^{3}) = 2 \times 3 = 6 \)