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) \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-\frac{dy}{dx}\left( x\frac{dy}{dx}-y \right)=0\]
E) \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+\left( x\frac{dy}{dx}-y \right)=0\]
Correct Answer: C
Solution :
Equation of family of ellipse is \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] On differentiating w.r.t.\[x,\]we get \[\Rightarrow \] \[\frac{2x}{{{a}^{2}}}+\frac{2y}{{{b}^{2}}}\frac{dy}{dx}=0\] \[\Rightarrow \] \[\frac{x}{{{a}^{2}}}+\frac{y}{{{b}^{2}}}\frac{dy}{dx}=0\] ?. (i) \[\Rightarrow \] \[\frac{y}{x}\frac{dy}{dx}=-\frac{{{b}^{2}}}{{{a}^{2}}}\] ?. (ii) Again differentiating w.r.t.\[x,\]we get \[\frac{1}{{{a}^{2}}}+\frac{y}{{{b}^{2}}}\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}\frac{1}{{{b}^{2}}}=0\] \[\Rightarrow \] \[\frac{{{b}^{2}}}{{{a}^{2}}}+y\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)+{{\left( \frac{dy}{dx} \right)}^{2}}=0\] \[\Rightarrow \] \[-\frac{y}{x}\frac{dy}{dx}+y\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}=0\] [from (ii)] \[\Rightarrow \] \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}\left( x\frac{dy}{dx}-y \right)=0\]
You need to login to perform this action.
You will be redirected in
3 sec
