A) \[\frac{{{x}^{3}}}{3{{y}^{3}}}+C\]
B) \[\frac{{{x}^{2}}}{{{y}^{2}}}+C\]
C) \[\frac{{{x}^{2}}}{3{{y}^{3}}}+C\]
D) \[\frac{{{x}^{3}}}{{{x}^{3}}+{{y}^{3}}}+C\]
Correct Answer: A
Solution :
Given, \[{{x}^{2}}y\,dx-({{x}^{3}}+{{y}^{3}})\,dy=0\] \[\Rightarrow \] \[\frac{dx}{dy}=\frac{{{x}^{3}}+{{y}^{3}}}{{{x}^{2}}y}\] ?.(i) Which is a homogeneous differential equation Now, put \[x=vy\] ?.(ii) On differentiation wrt to y, we get \[\frac{dx}{dy}=v.1+y\frac{dv}{dy}\] Then, from Eq. (i) \[v+y\frac{dv}{dy}=\frac{{{(vy)}^{3}}+{{y}^{3}}}{{{(vy)}^{2}}y}\] \[=\frac{{{v}^{3}}+1}{{{v}^{2}}}\] \[\Rightarrow \] \[v+y\frac{dv}{dy}=v+\frac{1}{{{v}^{2}}}\] \[\Rightarrow \] \[y\frac{dv}{dy}=\frac{1}{{{v}^{2}}}\] \[\Rightarrow \] \[\int{{{v}^{2}}\,dv=\int{\frac{dy}{y}}}\] (on integrating) \[\Rightarrow \] \[\frac{{{v}^{3}}}{3}=\log |y|-C\] \[\Rightarrow \] \[\log y=\frac{{{x}^{3}}}{3{{y}^{3}}}+C\] {from Eq. (ii)}\[\]
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