Formation of Ordinary Differential Equation by Elimination/Examples/y equals A e^2x + B e^-2x

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Examples of Formation of Ordinary Differential Equation by Elimination

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 $2$:

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


Proof

Differentiating twice with respect to $x$:

\(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds 2 A e^{2 x} - 2 B e^{-2 x}\) Derivative of Exponential Function
\(\ds \leadsto \ \ \) \(\ds \dfrac {\d^2 y} {\d x^2}\) \(=\) \(\ds 4 A e^{2 x} + 4 B e^{-2 x}\) Derivative of Exponential Function
\(\ds \) \(=\) \(\ds 4 y\) substituting from $(1)$

$\blacksquare$


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