The general solution of the differential equation \(x(1 + y^2)dx + y(1 + x^2)dy = 0\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \((1+ y^2) (1+x^2)= C\): where C is an arbitrary constant
The variables separate on dividing by \((1+x^2)(1+y^2)\):
\(\displaystyle \frac{x}{1+x^2}\,dx + \frac{y}{1+y^2}\,dy = 0\)
Integrating, using \(\displaystyle\int \frac{x}{1+x^2}\,dx = \frac{1}{2}\log(1+x^2)\), and writing the constant as \(\frac{1}{2}\log C\):
\(\displaystyle \frac{1}{2}\log(1+x^2) + \frac{1}{2}\log(1+y^2) = \frac{1}{2}\log C\)
\(\log\big[(1+x^2)(1+y^2)\big] = \log C\)
\((1+y^2)(1+x^2) = C\)
NCERT Exemplar:
Differential Equations
