A) \[{{x}^{3}}+{{y}^{2}}=p{{x}^{2}}\]
B) \[\frac{{{x}^{2}}}{2}+\frac{{{y}^{3}}}{x}={{y}^{2}}+p\]
C) \[{{x}^{2}}+{{y}^{3}}=p{{x}^{2}}\]
D) \[{{x}^{2}}+{{y}^{2}}=p{{x}^{3}}\]
Correct Answer: D
Solution :
It is homogeneous equation \[\frac{dy}{dx}=\frac{{{x}^{2}}+3{{y}^{2}}}{2xy}\] Put \[y=vx\] and \[\frac{dy}{dx}=v+x\frac{dv}{dx}\] So, we get \[x\frac{dv}{dx}=\frac{1+{{v}^{2}}}{2v}\] Þ \[\frac{2vdv}{1+{{v}^{2}}}=\frac{dx}{x}\] On integrating, we get \[{{x}^{2}}+{{y}^{2}}=p{{x}^{3}}\]. (where p is a constant)
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