Solution:

Given differential equation: xdx - dy = 0

To determine the type of curves formed by the solution to this differential equation, let's rearrange the equation:

xdx = dy

Now, we can integrate both sides of the equation with respect to their respective variables:

∫xdx = ∫dy

Integrating, we get:

(1/2)x^2 = y + C

where C is the constant of integration.

This is a second-degree polynomial equation, which represents a parabola. Therefore, the solution curves of the given differential equation form a family of parabolas.

Explanation:

- The given differential equation can be rearranged and integrated to obtain an equation of the form (1/2)x^2 = y + C, where C is a constant.
- This equation represents a parabola with its vertex at the point (0, C) and the axis of symmetry parallel to the y-axis.
- The constant C determines the position of the vertex and the shape of the parabola. Different values of C will result in different parabolas.
- As C varies, the parabolas obtained will form a family of curves that are distinct from each other but share the same general shape.
- Therefore, the solution curves of the given differential equation constitute a family of parabolas.
- It is important to note that the constant of integration C can take any value, leading to infinitely many parabolas in the family of solution curves.
- The parabolas can be open upwards or downwards, depending on the sign of the coefficient of x^2 in the differential equation.
- For example, if the differential equation is -xdx - dy = 0, the resulting parabolas would be open downwards.
- Similarly, if the differential equation is -xdx + dy = 0, the resulting parabolas would be open upwards.
- Therefore, the answer to the given question is option 'A' - parabolas.