A) \[x{{\left( \frac{dy}{dx} \right)}^{2}}-y\frac{dy}{dx}+1=0\]
B) \[x\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}-1=0\]
C) \[y\frac{dx}{dy}+x-1=0\]
D) \[y\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}=0\]
Correct Answer: A
Solution :
\[y-{{y}_{1}}=\frac{dy}{dx}(x-{{x}_{1}})\] \[P\left( 0,\,{{y}_{1}}-{{x}_{1}}\frac{dy}{dx} \right)\] \[\left( y-{{y}_{1}}+{{x}_{1}}\frac{dy}{dx} \right)=\frac{dx}{dy}(x)\] \[{{x}_{1}}{{\left( \frac{dy}{dx} \right)}^{2}}-{{y}_{1}}\frac{dy}{dx}+1=0\] As \[({{x}_{1}},\,{{y}_{1}})\] be any point, differential equation is given by \[x{{\left( \frac{dy}{dx} \right)}^{2}}-y\frac{dy}{dx}+1=0\].
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