A) \[\frac{1}{4}\left[ \cos \frac{x}{4}-\sin \frac{x}{4} \right]+C\]
B) \[4\left[ \cos \frac{x}{4}-\sin \frac{x}{4} \right]+C\]
C) \[4\left[ \sin \frac{x}{4}-\cos \frac{x}{4} \right]+C\]
D) \[4\left[ \sin \frac{x}{4}+\cos \frac{x}{4} \right]+C\]
Correct Answer: A
Solution :
\[({{x}^{2}}-y{{x}^{2}})\frac{dy}{dx}+{{y}^{2}}+x{{y}^{2}}=0\] \[\Rightarrow \] \[\frac{1-y}{{{y}^{2}}}dy+\frac{1+x}{{{x}^{2}}}dx=0\] \[\Rightarrow \]\[\left( \frac{1}{{{y}^{2}}}-\frac{1}{y} \right)dy+\left( \frac{1}{{{x}^{2}}}+\frac{1}{x} \right)dx=0\] On Integra mg, we get \[\log \left( \frac{x}{y} \right)=\frac{1}{x}+\frac{1}{y}+c\]
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