A) \[x\frac{dy}{dx}-y\]
B) \[{{\left( x\frac{dy}{dx}-y \right)}^{2}}\]
C) \[y\frac{dy}{dx}-x\]
D) \[{{\left( y\frac{dy}{dx}-x \right)}^{2}}\]
Correct Answer: B
Solution :
From the given relation \[\frac{y}{x}=\log x-\log (a+bx)\] Differentiating we get \[\frac{\left( x\frac{dy}{dx}-y \right)}{{{x}^{2}}}=\frac{1}{x}-\frac{1}{a+bx}b=\frac{a}{x(a+bx)}\] \[\therefore x\frac{dy}{dx}-y=\frac{ax}{a+bx}\] .....(i) Differentiating again w.r.t. x, we get \[x\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}-\frac{dy}{dx}=\frac{(a+bx)a-ax.b}{{{(a+bx)}^{2}}}\]Þ\[x\frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{{{a}^{2}}}{{{(a+bx)}^{2}}}\] Þ \[{{x}^{3}}\frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{{{a}^{2}}{{x}^{2}}}{{{(a+bx)}^{2}}}={{\left( x\frac{dy}{dx}-y \right)}^{2}}\] , [by (i)].
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