# Formation of Ordinary Differential Equation by Elimination/Examples/y equals A cos 3x + B sin 3x

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## Examples of Formation of Ordinary Differential Equation by Elimination

Consider the equation:

- $(1): \quad y = A \cos 3 x + B \sin 3 x$

This can be expressed as the ordinary differential equation of order $2$:

- $\dfrac {\d^2 y} {\d x^2} + 9 y = 0$

## Proof

Differentiating twice with respect to $x$:

\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds -3 A \sin 3 x + 3 B \cos 3 x\) | Derivative of Cosine Function, Derivative of Sine Function | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \dfrac {\d^2 y} {\d x^2}\) | \(=\) | \(\ds -9 A \cos 3 x - 9 B \sin 3 x\) | Derivative of Cosine Function, Derivative of Sine Function | ||||||||||

\(\ds \) | \(=\) | \(\ds -9 y\) | 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: $(2)$