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

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## 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

 $\ds \dfrac {\d y} {\d x}$ $=$ $\ds A$ 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 $(1)$

$\blacksquare$