The general solution of the differential equation \((1 + x^2)\,dy = x(1 + y)\,dx\) is (C is an arbitrary constant)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(1 + y = C\sqrt{1 + x^2}\)
The variables separate:
\( \displaystyle \frac{dy}{1 + y} = \frac{x\,dx}{1 + x^2} \)
On the right, the numerator x is half the derivative of \(1 + x^2\). Integrating both sides:
\( \displaystyle \log|1 + y| = \frac{1}{2}\log(1 + x^2) + \log|C| \)
\( \displaystyle \log|1 + y| = \log\left(|C|\sqrt{1 + x^2}\right) \)
\(1 + y = C\sqrt{1 + x^2}\)
Why the other options are incorrect
Option (2) \(1 + y = C(1 + x^2)\) — Incorrect. This misses the factor \(\frac{1}{2}\) in \(\int \frac{x}{1 + x^2}\,dx = \frac{1}{2}\log(1 + x^2)\).
Option (3) \((1 + y)\sqrt{1 + x^2} = C\) — Incorrect. This comes from a wrong sign; it solves \(\frac{dy}{dx} = -\frac{x(1 + y)}{1 + x^2}\).
Option (4) \((1 + y)^2 = x^2 + C\) — Incorrect. Differentiating gives \(\frac{dy}{dx} = \frac{x}{1 + y}\), a different equation.