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Formation of a Differential Equation from a General Solution

Form the diff...

Question

Form the differential equation corresponding to $y = A cos 3 x + B sin 3 x$ , where $A$ and $B$ are parameters.

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Solution

$y = A cos 3 x + B sin 3 x$ .... $(1)$
Differentiate
$y^{'} = - 3 A sin 3 x + 3 B cos 3 x$
again differentiate,
$y^{''} = - 9 A cos 3 x - 9 B sin 3 x$
$\Rightarrow y^{''} = - 9 (A cos 3 x + B sin 3 x) = - 9 y$ , using $(1)$
$\Rightarrow y^{''} + 9 y = 0$

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