Differential Equations

To find the differential equation representing a family of circles with centers on the $Y$-axis, let's consider the circle with center $(0, K)$ and radius $r$. The general equation for such a circle is:$x^2+(y-K)^2=r^2$Differentiate equation (i) with respect to $x$ :$2 x+2(y-K) \;\frac{d y}{d x}=0$Which simplifies to:$y-K=\frac{-x}{\frac{d y}{d x}} \;\; \text{ or } \;\; K=y+\frac{x}{\frac{d y}{d x}}$Let's denote $\;\frac{d y}{d x}$ as $y_1$. So, from the above, we have:$y-K=\frac{-x}{y_1} \;\; \Rightarrow \;\; K=y+\frac{x}{y_1}$Now, differentiate this equation with respect to $x$ again:$0=1+(y-K) \;\frac{d^2 y}{d x^2}+ \left(\frac{d y}{d x}\right)^2$Substituting $y-K=\frac{-x}{y_1}$ into this, we get:$1+ \left(-\frac{x}{y_1}\right) \;\frac{d^2 y}{d x^2}+y_1^2=0$This can be rearranged to:$\;1-\frac{x \frac{d^2 y}{d x^2}}{y_1}+y_1^2=0$$\;x \;\frac{d^2 y}{d x^2}=y_1 (y_1^2+1)$So, the differential equation is:$x \;\frac{d^2 y}{d x^2}=y_1 (y_1^2+1)$