Formation of Ordinary Differential Equation by Elimination/Examples/Straight Line through Origin

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Examples of Formation of Ordinary Differential Equation by Elimination

Consider the set of all straight lines embedded in the Cartesian plane which pass through the origin.

This set can be expressed as the ordinary differential equation of order $1$:

$\dfrac y x = \dfrac {\d y} {\d x}$

That is, the tangent at any point on a straight line through the origin is the straight line itself.


From Equation of Straight Line in Plane: Slope-Intercept Form, such a straight line has the equation:

$y = m x$

Differentiating with respect to $x$:

\(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds m\) Power Rule for Derivatives
\(\ds \leadsto \ \ \) \(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds \dfrac y x\)

Hence the result.