A) \[224{{y}^{2}}\]
B) \[125y\]
C) \[225y\]
D) \[225{{y}^{2}}\]
Correct Answer: C
Solution :
\[y=\left\{ x+{{\sqrt{{{x}^{2}}-1}}^{15}} \right\}+{{\left\{ x-\sqrt{{{x}^{2}}-1} \right\}}^{15}}\] \[\frac{dy}{dx}=15{{\left( x+\sqrt{{{x}^{2}}-1} \right)}^{14}}15{{\left( x-\sqrt{{{x}^{2}}-1} \right)}^{14}}\left( 1-\frac{x}{\sqrt{{{x}^{2}}-1}} \right)\] \[\frac{dy}{dx}=\frac{15}{\sqrt{{{x}^{2}}-1}}.y\] ?(i) \[\sqrt{{{x}^{2}}-1}.\frac{dy}{dx}=15y\] \[\frac{x}{\sqrt{{{x}^{2}}-1}}.\frac{dy}{dx}+\sqrt{{{x}^{2}}-1}\frac{{{d}^{2}}y}{d{{x}^{2}}}=15\frac{dy}{dx}\] \[x\frac{dy}{dx}+\left( {{x}^{2}}-1 \right)\frac{{{d}^{2}}y}{d{{x}^{2}}}=15\sqrt{{{x}^{2}}-1}.\frac{15}{\sqrt{{{x}^{2}}-1}}.y=225y\]
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