If \(y = y(x)\), \(x \gt 0\), is the solution of the differential equation \(x\displaystyle\frac{dy}{dx} = y\log_e y\) satisfying \(y(1) = e^{2}\), then \(y\left(\displaystyle\frac{3}{2}\right)\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(e^{3}\)
The equation is variables separable (here \(y \gt 1\), since y(1) = \(e^2\), so \(\log_e y \gt 0\)):
\( \displaystyle \frac{dy}{y\log_e y} = \frac{dx}{x} \)
On the left, \(\frac{1}{y}\) is the derivative of \(\log_e y\). Putting \(t = \log_e y\), \(dt = \frac{dy}{y}\):
\( \displaystyle \int \frac{dt}{t} = \int \frac{dx}{x} \)
\(\log_e t = \log_e x + \log_e C\)
\(t = Cx\)
So the general solution is
\(\log_e y = Cx\)
Putting x = 1, y = \(e^2\):
\(2 = C\)
\(\log_e y = 2x\)
At \(x = \frac{3}{2}\):
\(\log_e y = 3\)
\(y = e^{3}\)