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
- 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: $(4)$