If y = y(x) is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = \frac{x\left(1 + y^2\right)}{y\left(1 + x^2\right)}\) with y(0) = 1 and y > 0, then \(y\left(2\sqrt{3}\right)\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 5
The variables separate:
\( \displaystyle \frac{y\,dy}{1 + y^2} = \frac{x\,dx}{1 + x^2} \)
Each side is half the derivative of a logarithm, so integrating:
\( \displaystyle \frac{1}{2}\log\left(1 + y^2\right) = \frac{1}{2}\log\left(1 + x^2\right) + c \)
\(1 + y^2 = A\left(1 + x^2\right)\)
With y(0) = 1:
\(2 = A\)
\(1 + y^2 = 2\left(1 + x^2\right)\)
\(y^2 = 1 + 2x^2\)
At \(x = 2\sqrt{3}\), \(x^2 = 12\):
\(y^2 = 1 + 24 = 25\)
\(y = 5\)
Why the other options are incorrect
Option (2) \(\sqrt{13}\) — Incorrect. This comes from adding the constant (\(1 + y^2 = 1 + x^2 + 1\)); the constant multiplies \(1 + x^2\).
Option (3) 7 — Incorrect. \(7^2 = 49\), but \(y^2 = 25\).
Option (4) 25 — Incorrect. This is the value of \(y^2\), not of y.