Formation of a Differential Equation

So far you've learned what a differential equation is and what its solution looks like. Now turn the process around: given a family of curves — infinitely many curves sharing the same shape but differing in the values of one or more parameters — can you find the single differential equation that every member of that family satisfies?

Take the family of all circles centred at a fixed point, say $(2, 3)$, with a fixed radius, say $5$. Differentiating the equation of this circle once with respect to $x$ eliminates the centre and radius from the equation and leaves behind a differential equation — one that every circle sharing that centre and radius satisfies.

Now take the family of straight lines $y = mx + c$, where $m$ (the slope) and $c$ (the $y$-intercept) are parameters: choosing different values of $m$ and $c$ traces out different members of the family. Since this family has two arbitrary constants, differentiating once is not enough to eliminate both. Differentiating $y = mx + c$ once removes $c$ but leaves $m$ behind; differentiating a second time removes $m$ as well, giving