Find the differential equation of the family of curves represented by y^{2}= a\left ( b^{2}-x^{2} \right )\cdot

 

 

 

 
 
 
 
 

Answers (1)

y^{2}= a\left ( b^{2}-x^{2} \right )\; \; y^{2}= ab^{2}-ax^{2}
   \Rightarrow 2y{y}'= 0- 2ax
  \Rightarrow \frac{y{y}'}{x}= -a
\Rightarrow \frac{x\times \left ( y{y}'' +{y}'{y}'\right )-y{y}'(1)}{x^{2}}= 0
\therefore xy\frac{d^{2}y}{dx^{2}}+x\left ( \frac{dy}{dx} \right )^{2}-y\left ( \frac{dy}{dx} \right )= 0

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