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Q.
The differential equation of the family of circles touching the $x$-axis at origin is given by
Differential Equations
Solution:
Family of circles represented graphically is shown below.
Equation of the family of circles is
$x^2 + (y - a)^2 = a^2$
$\Rightarrow x^{2}+y^{2}=2ay\quad\ldots\left(i\right)$
Differentiating $(i)$ w.r.t. $x$, we get
$2x+2y \frac{dy}{dx}=2a \frac{dy}{dx}$
$\Rightarrow x+yy'=ay'$
$\Rightarrow a=\frac{x+yy'}{y'}$
Substituting value of $‘a’$ in equation $(i)$
$x^{2}+y^{2}=2y\left(\frac{x+yy'}{y'}\right)$
$\Rightarrow x^{2}\,y'+y^{2}\,y'=2xy+2y^{2}\,y'$
$\Rightarrow y'=\frac{2xy}{x^{2}-y^{2}}$
