I'll extract and analyze the mathematical equations from your uploaded image, then solve the equations step by step.

Here are the extracted equations:

1. $\tan(y) \, dx - \cot(x) \, dy = 0$
2. $(12x + 5y - 9) \, dx + (5x + 2y - 3) \, dy = 0$
3. $x \frac{dy}{dx} = y + \sqrt{x^2 + y^2}$
4. $x \frac{dy}{dx} + y + x^3 = 0$
5. $y \, dx - x \, dy = x^2 y \, dy$
6. ( \frac{dx}{dt} + 3x = e^{2