If \(y = y(x)\) is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = \frac{y^2}{x}\), \(x \gt 0\), satisfying \(y(1) = 1\), then \(y\left(\sqrt{e}\right)\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 2
The variables separate:
\( \displaystyle \frac{dy}{y^2} = \frac{dx}{x} \)
Integrating both sides:
\( \displaystyle -\frac{1}{y} = \log x + C \)
Putting x = 1, y = 1:
\(-1 = 0 + C,\quad C = -1\)
\( \displaystyle \frac{1}{y} = 1 - \log x \)
At \(x = \sqrt{e}\), \(\log\sqrt{e} = \frac{1}{2}\):
\( \displaystyle \frac{1}{y} = 1 - \frac{1}{2} = \frac{1}{2} \)
\(y = 2\)