The general solution of the differential equation \( x\,dx=(x^2+1)\,dy \) is (where \( C \) is an arbitrary constant)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( y=\displaystyle\frac{1}{2}\log_e|x^2+1|+C \)
The variables separate at once:
\( \displaystyle dy=\frac{x}{x^2+1}\,dx \)
To integrate the right side, put \( x^2+1=t \), so \( 2x\,dx=dt \):
\( \displaystyle \int\frac{x}{x^2+1}\,dx=\frac{1}{2}\int\frac{dt}{t} \)
\( \displaystyle =\frac{1}{2}\log_e|t| \)
\( \displaystyle y=\frac{1}{2}\log_e|x^2+1|+C \)
Why the other options are incorrect
Option (1) \( y=\log_e|x^2+1|+C \) — Incorrect. The factor \( \frac{1}{2} \) from \( x\,dx=\frac{1}{2}\,dt \) has been dropped.
Option (3) \( \displaystyle\frac{y^2}{2}=\displaystyle\frac{x^3}{3}+x+C \) — Incorrect. This comes from \( y\,dy=(x^2+1)\,dx \), with the variables x and y interchanged.
Option (4) \( y=\displaystyle\frac{x^2}{2}+\log_e|x|+C \) — Incorrect. This comes from \( dy=\frac{x^2+1}{x}\,dx \), with the fraction turned upside down.