Steps to Form a Differential Equation

When a family of curves depends on $n$ independent parameters, forming its differential equation is a systematic elimination process:

1. Write the equation of the family in the form $f(x, y, a_1, a_2, \ldots, a_n) = 0$, containing all $n$ parameters.
2. Differentiate this equation with respect to $x$. If $n = 1$, this single differentiation already gives an equation free of the parameter — combine it with the original equation to eliminate $a_1$, and you are done.
3. If $n \geq 2$, one differentiation still carries the remaining parameters. Differentiate again (and again, as needed) until you have exactly $n$ equations in total, including the original.
4. Eliminate all $n$ parameters from these $n$ equations. What remains, free of every parameter, is the required differential equation. …