Any line passing through origin is given by
y = mx
on differentiating the above equation to work with respect to x
y' = m
So, the differential equation is,
y = xy' (or) y' = y/x

Differential Equation of Straight Lines Passing Through the Origin

To find the differential equation of all straight lines passing through the origin, we first consider the general equation of a straight line passing through the origin: y = mx.

Derivation:

Let y = mx be the equation of a straight line passing through the origin.
Differentiating both sides with respect to x, we get:
dy/dx = m

This is the first derivative of y with respect to x, which represents the slope of the line. Since the line passes through the origin, the y-intercept is 0.

General Form of Differential Equation:

The differential equation of all straight lines passing through the origin can be written as:
dy/dx = y/x

This equation represents the relationship between the first derivative of y with respect to x and the ratio of y to x, which is the slope of the line passing through the origin.

Conclusion:

The differential equation of all straight lines passing through the origin is dy/dx = y/x. This equation captures the essential relationship between the slope of the line and the coordinates of points lying on the line, providing a mathematical description of straight lines passing through the origin.