The general solution of the differential equation \((x^2-yx^2)\,dy + (y^2+x^2y^2)\,dx = 0\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\log_e|y|+\displaystyle\frac{1}{x}+\frac{1}{y}-x= c\), where \(c\) is constant of integration
Taking out the common factors shows that the variables are separable:
\(x^2(1-y)\,dy + y^2(1+x^2)\,dx = 0\)
Dividing by \(x^2y^2\):
\( \displaystyle \frac{1-y}{y^2}\,dy + \frac{1+x^2}{x^2}\,dx = 0 \)
\( \displaystyle \left(\frac{1}{y^2} - \frac{1}{y}\right) dy + \left(\frac{1}{x^2} + 1\right) dx = 0 \)
Integrating term by term, with k the constant:
\( \displaystyle -\frac{1}{y} - \log_e|y| - \frac{1}{x} + x = k \)
Multiplying by \(-1\) and writing \(c = -k\):
\( \displaystyle \log_e|y| + \frac{1}{x} + \frac{1}{y} - x = c \)