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 $(1)$ |
$\blacksquare$
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: $(1)$