The solution of differential equation \(y\left(1 - x^2\right)\displaystyle\frac{dy}{dx} = x\left(1 + y^2\right)\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\left(1 + y^2\right)\left|1 - x^2\right| = C\)
The equation is variables separable:
\( \displaystyle \frac{y}{1 + y^2}\,dy = \frac{x}{1 - x^2}\,dx \)
In each integral the numerator is half the derivative of the denominator:
\( \displaystyle \int\frac{y}{1 + y^2}\,dy = \frac{1}{2}\log\left(1 + y^2\right) \)
\( \displaystyle \int\frac{x}{1 - x^2}\,dx = -\frac{1}{2}\log\left|1 - x^2\right| \)
So
\( \displaystyle \frac{1}{2}\log\left(1 + y^2\right) = -\frac{1}{2}\log\left|1 - x^2\right| + c \)
\( \log\left(1 + y^2\right) + \log\left|1 - x^2\right| = 2c \)
\( \log\left[\left(1 + y^2\right)\left|1 - x^2\right|\right] = 2c \)
\( \left(1 + y^2\right)\left|1 - x^2\right| = e^{2c} = C \)