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

## 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 $(1)$

$\blacksquare$