A) \[\frac{1}{{{x}^{2}}-1}\]
B) \[{{x}^{2}}-1\]
C) \[\frac{{{x}^{2}}-1}{x}\]
D) \[\frac{x}{{{x}^{2}}-1}\]
Correct Answer: B
Solution :
Given differential equation is \[\left( {{x}^{2}}-1 \right)\frac{dy}{dx}+2xy=x\]\[\Rightarrow \]\[\frac{dy}{dx}+\frac{2x}{{{x}^{2}}-1}.y=\frac{x}{{{x}^{2}}-1}\] This is in linear form. Integrating factor \[\int\limits_{e}^{{}}{\frac{2x}{{{x}^{2}}-1}}dx=\int\limits_{e}^{{}}{\frac{dt}{t}}\]where\[t={{x}^{2}}-1\]\[={{e}^{\log t}}={{x}^{2}}-1\] Hence, required integrating factor \[={{x}^{2}}-1.\]
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