If \(y = y(x)\) is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = \frac{2}{x}\), \(x \gt 0\), satisfying \(y(1) = 3\), then \(y(e^2)\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 7
The right side contains only x, so the solution comes by direct integration:
\(y = 2\log x + C\)
Putting x = 1, y = 3, and using \(\log 1 = 0\):
\(3 = 0 + C,\quad C = 3\)
\(y = 2\log x + 3\)
At \(x = e^2\), \(\log e^2 = 2\):
\(y(e^2) = 2 \times 2 + 3 = 7\)