To find the differential equation of the family of lines passing through the origin, let's start by considering the general equation of a line passing through the origin in slope-intercept form:
\(y = mx\)
where \(m\) is the slope of the line.
To find the differential equation, we need to eliminate the parameter \(m\). To achieve this, we differentiate the equation with respect to \(x\):
\(\frac{dy}{dx} = m\)
From the original equation \(y = mx\), we can express \(m\) as:
\(m = \frac{dy}{dx}\)
Substitute this expression for \(m\) back into the original equation:
\(y = \left(\frac{dy}{dx}\right) \cdot x\)
Rearrange the above equation to form a differential equation:
\(x \frac{dy}{dx} - y = 0\)
The obtained equation \(x \frac{dy}{dx} - y = 0\) is the differential equation for the family of lines that pass through the origin.
Thus, the correct answer is \(x \frac{dy}{dx} - y = 0\), which corresponds to the third option provided.