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 $(1)$

$\blacksquare$

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: $(2)$