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Q.
The differential equation of the family of circles having centre on $Y$-axis and radius 3 units is
Differential Equations
Solution:
The equation of the family of circle having centre on $y$-axis and radius 3 units is
$x^2+(y-b)^2=9 .....$(i)
On differentiating Eq.(i) w.r.t. $x$, we get
$2 x+2(y-b) y^{\prime}=0$
$\Rightarrow y-b=-\frac{x}{y^{\prime}} ....$(ii)
On substituting this value of $(y-b)$ in Eq. (i), we get
$x^2+\left(-\frac{x}{y^{\prime}}\right)^2 =9 $
$\Rightarrow x^2\left[\left(y^{\prime}\right)^2+1\right] =9\left(y^{\prime}\right)^2 $
$\Rightarrow \left(x^2-9\right)\left(y^{\prime}\right)^2+x^2 =0$
which is the required differential equation.
