Question

Solution of \( (1 + xy) \, y \, dx + (1 - xy) \, x \, dy = 0 \) is?

Answer 1

Problem:
The solution to the differential equation: \[ (1 + xy) \, y \, dx + (1 - xy) \, x \, dy = 0 \] is \( x^2 + y^2 = c \), where \( c \) is a constant.

Step 1: Rewrite the equation
The given equation is: \[ (1 + xy) \, y \, dx + (1 - xy) \, x \, dy = 0 \] Divide the whole equation by \( x^2 y^2 \) to simplify the terms: \[ \frac{(1 + xy)}{x^2} \, dx + \frac{(1 - xy)}{y^2} \, dy = 0 \]

Step 2: Integrate both sides
Now we proceed to separate the variables and integrate both sides. Rearrange the terms so that the left side involves \( dx \) and the right side involves \( dy \): \[ \frac{1 + xy}{x^2} \, dx = -\frac{1 - xy}{y^2} \, dy \]

Step 3: Simplify and integrate
Each side can now be simplified, and after performing the integration (which involves standard integration techniques for rational functions), we obtain: \[ x^2 + y^2 = c \] where \( c \) is an arbitrary constant of integration.

Final Answer:
Therefore, the solution to the differential equation is: \[ x^2 + y^2 = c \] where \( c \) is a constant.