Form the differential equation representing the family of curves y^{2}= m\left ( a^{2}-x^{2} \right ) by eliminating the arbitrary constants 'm' and 'a'.

 

 

 

 
 
 
 
 

Answers (1)

y^{2}= m\left ( a^{2}-x^{2} \right ) is the given equation where m and a are arbitary constants
 y^{2}= m\left ( a^{2}-x^{2} \right ) ---\left ( i \right )
\frac{2ydy}{dx}= -2mx---\left ( ii \right )
\Rightarrow -2m= \frac{2y}{x}\frac{dy}{dx}
2\left [ y\frac{d^{2}y}{dx^{2}}+\left ( \frac{dy}{dx} \right )^{2} \right ]= -2m---\left ( iii \right )
2\left [ y\frac{d^{2}y}{dx^{2}}+\left ( \frac{dy}{dx} \right )^{2} \right ]= 2\frac{y}{x}\frac{dy}{dx}    y\frac{d^{2}y}{dx^{2}}+\left ( \frac{dy}{dx} \right )^{2}-\left ( \frac{y}{x} \right )\frac{dy}{dx}= 0
\therefore The above is the required differential equation

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