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  • Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b. y^2 = a (b^2 - x^2)

2. Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

       y^2 = a(b^2 - x^2)

Answers (1)

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Given equation is
y^2 = a(b^2 - x^2)                      
Differentiate both the sides w.r.t x
\frac{d\left ( y^2 \right )}{dx}=\frac{d(a(b^2-x^2))}{dx}
2y\frac{dy}{dx}= -2ax\\ \\ y.\frac{dy}{dx}= -ax\\ \\ y.y^{'}=-ax                  -(i)
Now, again differentiate it w.r.t x
y^{'}.y^{'}+y.y^{''}= -a\\ (y^{'})^2+y.y^{''}=-a                 -(ii)
Now, divide equation (i) and (ii)
\frac{(y^{'})^2+y.y^{''}}{y.y^{'}}= \frac{-a}{-ax}\\ \\ x(y^{'})^2+x.y.y^{''}=y.y^{'}\\ \\ x(y^{'})^2+x.y.y^{''}-y.y^{'}=0
Therefore, the required differential equation is  x(y^{'})^2+x.y.y^{''}-y.y^{'}=0

Posted by

Gautam harsolia

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