The general solution of the differential equation \(\displaystyle\frac{dy}{dx}= xy + x + y + 1\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\log_e|y+1| = \displaystyle\frac{1}{2}x^2+x+C\), where C is constant of integration
The right side factorises, so the variables separate:
\(xy + x + y + 1 = x(y + 1) + (y + 1) = (x + 1)(y + 1)\)
\( \displaystyle \frac{dy}{y + 1} = (x + 1)\,dx \)
Integrating both sides:
\( \displaystyle \int \frac{dy}{y + 1} = \int (x + 1)\,dx \)
\( \displaystyle \log_e|y + 1| = \frac{1}{2}x^2 + x + C \)