Explanation:

To determine the type of curve represented by the equation xdy = ydx, we can rearrange the equation by dividing both sides by xy:

dy/dx = y/x

This is a first-order linear homogeneous ordinary differential equation (ODE). By rewriting it in the form:

dy/dx - (y/x) = 0

We can see that it is in the standard form of a linear ODE:

dy/dx + P(x)y = Q(x)

where P(x) = -1/x and Q(x) = 0.

Linear Ordinary Differential Equations:

Linear ODEs have the general form:

dy/dx + P(x)y = Q(x)

where P(x) and Q(x) are functions of x.

The general solution to a linear ODE is given by:

y = e^(-∫P(x)dx) * (∫Q(x) * e^(∫P(x)dx) * dx + C)

where C is the constant of integration.

Solving the Equation:

In the case of the given equation dy/dx - (y/x) = 0, we have P(x) = -1/x and Q(x) = 0.

Integrating P(x), we get:

∫P(x)dx = ∫(-1/x)dx = -ln|x| + C1

where C1 is the constant of integration.

Substituting the value of ∫P(x)dx into the general solution, we have:

y = e^(-(-ln|x| + C1)) * (∫0 * e^(-ln|x| + C1) * dx + C2)
= e^(ln|x| - C1) * (C2)
= |x| * C2

where C2 is another constant of integration.

Conclusion:

The general solution y = |x| * C2 represents a family of straight lines passing through the origin (0,0). Each value of C2 corresponds to a different straight line in the family.

Therefore, the equation xdy = ydx represents a family of straight lines.