The equation of families of straight lines passing through origin is y = ax, where a is arbitrary constant.
y = ax ....(1)
Differentiate equation (1) w.r.t x, we get
\(\frac{dy}{dx} = a\)
put \(a = \frac{dy}{dx}\) in equation (1), we get
\(y = x\frac{dy}{dx}\)
\(\Rightarrow x\frac{dy}{dx} - y = 0 ....(2)\)
Equation (2) represent the differential equation of the family of lines through the origin.