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

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