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Equations Involving sinx + cosx and sinx.cosx

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Question

If the differential equation satisfied by $y = A sin (98 t + w_{0}) + B cos (98 t + w_{0})$ ( $w_{0}$ is a known constant) is $y_{2} + C y = 0$ then the value of C is

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Solution

$\frac{d y}{d t} = 98 A cos (98 t + w_{0}) - 98 B sin (98 + w_{0})$
$\Rightarrow \frac{d^{2} y}{d t^{2}} = - 9604 A sin (98 t + w_{0}) - 9604 B cos (98 t + w_{0}) = - 9604 y$
$\Rightarrow y_{2} + 9604 y = 0$ .
Hence $A = 9604$ .

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