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Find the general solution of the differential equation x*dy + y*dx = e^(x*y - ln(x²)) * (x*dy - y*dx), where C is an arbitrary constant.

1. y/x + e^(-x*y) = C
2. x/y + e^(-x*y) = C
3. -y/x + e^(-x*y) = C
4. -x/y + e^(-x*y) = C

Correct answer: y/x + e^(-x*y) = C

Solution

Rewrite the equation: d(xy) = (e^(xy)/x²) * x² * d(y/x). Substituting t = y/x: d(xy) = e^(xy) * d(t). Now xy = x*(x*t) = t*x². Let u = xy. Then du = e^u * dt => e^(-u) du = dt. Integrating: -e^(-u) = t + K => -e^(-xy) = y/x + K => y/x + e^(-xy) = C (where C = -K).

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