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