The general solution of the differential equation \((1+x^2)\,dy-x\,dx=0\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{e^{2y}}{1+x^2}=C\)
Separating the variables:
\( \displaystyle dy = \frac{x}{1+x^2}\,dx \)
Integrating both sides (the numerator x is half the derivative of \(1+x^2\)):
\( \displaystyle y = \frac{1}{2}\log(1+x^2)+c \)
\( 2y = \log(1+x^2)+2c \)
\( e^{2y} = e^{2c}(1+x^2) \)
Writing \(C = e^{2c}\):
\( \displaystyle \frac{e^{2y}}{1+x^2} = C \)