If \(y = y(x)\) is the solution of the differential equation \((1 + x^2)\displaystyle\frac{dy}{dx} = 1\) satisfying \(y(0) = 1\), then \(y(\sqrt{3})\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(1 + \displaystyle\frac{\pi}{3}\)
Separating the variables:
\( \displaystyle dy = \frac{dx}{1 + x^2} \)
Integrating both sides:
\(y = \tan^{-1} x + C\)
Putting x = 0, y = 1 gives C = 1, so the particular solution is
\(y = \tan^{-1} x + 1\)
At \(x = \sqrt{3}\), where \(\tan^{-1}\sqrt{3} = \frac{\pi}{3}\):
\( \displaystyle y(\sqrt{3}) = 1 + \frac{\pi}{3} \)