Find the equation of a curve which passes through the origin and whose differential equation is

Solve (1 + x²) sec2 y dy + 2x tan y dx = 0, given that when x = 1.

Find the equation of the curve passing through the point whose differential equation is sin x cos y dx + cos x sin y dy = 0.

A curve passes through the point (0, -2) and at any point (x, y) of the curve, the product of the slope of its tangent and y-coordinate of the point is equal to the x-coordinate of the point. Find the equation of the curve.

A curve passes through the point (-1, 1) and at any point (x, y) of the curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (-4, -3). Find the equation of the curve.