# Formation of Ordinary Differential Equation by Elimination/Examples/y equals Ax + A^3

Jump to navigation
Jump to search

## Examples of Formation of Ordinary Differential Equation by Elimination

Consider the equation:

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

This can be expressed as the ordinary differential equation:

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

## Proof

Differentiating with respect to $x$:

\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds A\) | Power Rule for Derivatives | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds x \dfrac {\d y} {\d x} + \paren {\dfrac {\d y} {\d x} }^3\) | substituting for $A$ in $(1)$ |

$\blacksquare$

## 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: $(4)$