Formation of Ordinary Differential Equation by Elimination/Examples

Examples of Formation of Ordinary Differential Equation by Elimination

Simple Harmonic Motion

Consider the equation governing simple harmonic motion:

$(1): \quad y = A \map \cos {\omega x + \phi}$

This can be expressed as the ordinary differential equation of order $3$:

$y \cdot \dfrac {\d^3 y} {\d x^3} = \dfrac {\d y} {\d x} \cdot \dfrac {\d^2 y} {\d x^2}$

Parabolas whose Axes are $x$-Axis

Consider the set of all parabolas embedded in the Cartesian plane whose axis is the $x$ axis.

This set can be expressed as the ordinary differential equation of order $2$:

$y \dfrac {\d^2 y} {\d x^2} + \paren {\dfrac {\d y} {\d x} }^2 = 0$

Straight Line through Origin

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.

Example: $y = A e^{2 x} + B e^{-2 x}$

Consider the equation:

$(1): \quad y = A e^{2 x} + B e^{-2 x}$

This can be expressed as the ordinary differential equation of order $2$:

$\dfrac {\d^2 y} {\d x^2} = 4 y$

Example: $y = A \cos 3 x + B \sin 3 x$

Consider the equation:

$(1): \quad y = A \cos 3 x + B \sin 3 x$

This can be expressed as the ordinary differential equation of order $2$:

$\dfrac {\d^2 y} {\d x^2} + 9 y = 0$

Example: $y = A e^{B x}$

Consider the equation:

$(1): \quad y = A e^{B x}$

This can be expressed as the ordinary differential equation of order $2$:

$\dfrac {\d^2 y} {\d x^2} + 9 y = 0$

Example: $y = Ax + A^3$

Consider the equation:

$(1): \quad y = A x + A^3$

This can be expressed as the ordinary differential equation:

$y = x \dfrac {\d y} {\d x} + \paren {\dfrac {\d y} {\d x} }^3$

Example: $x^2 + y^2 = a^2$

Consider the equation:

$(1): \quad x^2 + y^2 = a^2$

This can be expressed as the ordinary differential equation:

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

which demonstrates that the radius of a circle where it meets the circle is perpendicular to the tangent at that point.