A) \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+10\frac{dy}{dx}+25y=0\]
B) \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-10\frac{dy}{dx}+25y=0\]
C) \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-25y=0\]
D) \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+25y=0\]
E) none of the above
Correct Answer: D
Solution :
\[y=-A\cos 5x+B\sin 5x\] \[\frac{dy}{dx}=5A\sin 5x+5Bcos5x\] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=25A\cos 5x-25B\sin 5x\] \[\Rightarrow \] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=-25[-A\cos 5x+B\sin 5x]=-25y\] \[\Rightarrow \] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+25y=0\]
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