A) \[2\cos \,(xy)+{{x}^{-2}}=C\]
B) \[2\cos \,(xy)-{{y}^{-2}}=C\]
C) \[2\sin y+{{x}^{-3}}=C\]
D) \[2\sin (xy)+{{y}^{-2}}=C\]
Correct Answer: A
Solution :
[a] \[{{x}^{4}}dy+{{x}^{3}}ydx+\cos ec(xy)dx=0\] \[\Rightarrow \,\,\,{{x}^{3}}(x\,\,dy+\,\,y\,\,dx)+\cos ec(xy)dx=0\] \[\Rightarrow \,\,\,\int{\frac{d(xy)}{\cos ec\,(xy)}}+\int{\frac{dx}{{{x}^{3}}}}=0\] \[\Rightarrow \,\,\,-\cos \ \,xy+\frac{{{x}^{-3+1}}}{-3+1}=C\] Thus, \[2\cos \,\,(xy)+{{x}^{-2}}=C\] is the general solution of given differential equation.
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