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

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.

Proof

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.

$\blacksquare$

Sources

• 1952: H.T.H. Piaggio: An Elementary Treatise on Differential Equations and their Applications (revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation: Examples for solution: $(6)$