A) \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}\left( x\frac{dy}{dx}-y \right)=0\]
B) \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}-\frac{dy}{dx}\left( x\frac{dy}{dx}-y \right)=0\]
C) \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}\left( x\frac{dy}{dx}-y \right)=0\]
D) \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-\frac{dy}{dx}\left( x\frac{dy}{dx}-y \right)=0\]
Correct Answer: C
Solution :
If major axis is \[y-\]axis, then equation of ellipse is\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1,\,\,b>a\]. On differentiating w.r.t.\[x,\] we get \[\frac{2x}{{{a}^{2}}}+\frac{2y}{{{b}^{2}}}\frac{dy}{dx}=0\] ? (i) On again differentiating; w.r.t.\[x,\] we get \[\frac{2}{{{a}^{2}}}+\frac{2}{{{b}^{2}}}\left[ {{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{{{d}^{2}}y}{d{{x}^{2}}} \right]=0\] \[\Rightarrow \] \[\frac{{{b}^{2}}}{{{a}^{2}}}+\left[ {{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{{{d}^{2}}y}{d{{x}^{2}}} \right]=0\] From Eq. (i)\[\frac{{{b}^{2}}}{{{a}^{2}}}=-\frac{y}{x}\frac{dy}{dx}\] \[\therefore \] \[-\frac{y}{x}\frac{dy}{dx}+\left[ {{\left( \frac{dy}{dx} \right)}^{2}}+\frac{{{d}^{2}}y}{d{{x}^{2}}} \right]=0\] \[\Rightarrow \] \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}\left( x\frac{dy}{dx}-y \right)=0\] Note: The differential equation of family of ellipse is does not affect, either the major axis is on\[x-\]axis or\[y-\]axis.
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