A) a system of parabolas
B) a system of circles
C) \[{{y}^{2}}=x(1+x)-1\]
D) \[{{(x-2)}^{2}}+{{(y-3)}^{2}}=5\]
Correct Answer: C
Solution :
[c] Rewriting the given equation as \[2xy\frac{dy}{dx}-{{y}^{2}}=1+{{x}^{2}}\] Or \[2y\frac{dy}{dx}-\frac{1}{x}{{y}^{2}}=\frac{1}{x}+x\] Putting \[{{y}^{2}}=u,\]we have \[\frac{du}{dx}-\frac{1}{x}u=\frac{1}{x}+x\] I.F. \[={{e}^{-\int{\frac{1}{x}dx}}}=\frac{1}{x}\] Thus, solution is u \[\frac{1}{x}=\int{\left( \frac{1}{{{x}^{2}}}+1 \right)}dx=-\frac{1}{x}+x+C\] Or \[{{y}^{2}}=({{x}^{2}}-1)+Cx\] Since y(1) =1, we get C=1. Hence, \[{{y}^{2}}=x(1+x)-1\] which represents a systems of hyperbola.
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