A) \[A{{\omega }^{5}}\cos \left( \omega t-\frac{\pi }{2} \right)\]
B) \[A{{\omega }^{5}}\sin \left( \omega t-\frac{\pi }{2} \right)\]
C) \[A{{\omega }^{5}}\cos \left( \omega t+\frac{\pi }{2} \right)\]
D) \[A{{\omega }^{5}}\sin \left( \omega t+\frac{\pi }{2} \right)\]
Correct Answer: D
Solution :
\[y=A\sin \omega t\]. \[\therefore \,\,\frac{dy}{dx}=A\omega \cos \omega t\] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=-A{{\omega }^{2}}\sin \omega t\] \[\frac{{{d}^{3}}y}{d{{x}^{3}}}=A{{\omega }^{3}}\cos \omega t\] \[\frac{{{d}^{4}}y}{d{{x}^{4}}}=+A{{\omega }^{4}}\sin \omega t\] \[\therefore \] \[\frac{{{d}^{5}}y}{d{{x}^{5}}}=A{{\omega }^{5}}\cos \omega t=A{{\omega }^{5}}\sin \left( \omega t+\frac{\pi }{2} \right)\]
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