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Q.
The differential equation of the family of circles with fixed radius $r$ and with centre on $y-axis$ is
Differential Equations
Solution:
The equation of the family of circles with fixed radius $r$ and centre on $y$-axis is
$x^{2} + \left(y - K\right)^{2} = r^{2}\quad\ldots\left(1\right)$
where $K$ is a parameter.
Diff. $\left(1\right)$ both sides w.r.t. $‘x’$, we get
$2x+2\left(y-K\right) \frac{dy}{dx} = 0$
i.e., $x + \left(y -K \right) \frac{dy}{dx} = 0\quad\ldots\left(2\right)$
$\therefore y - K = -\frac{x}{y_{1}}\quad\ldots\left(3\right)$
Putting in $\left(1\right)$, we get $\left[\text{Where} \,y_{1} = \frac{dy}{dx}\right]$
$x^{2} + \frac{x^{2}}{y^{2}_{1}} = r^{2}$
$\Rightarrow x^{2}\left(1+y^{2}_{1}\right) = r^{2}y^{2}_{1}$