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Formation of Differential Equations

Consider a family of curves
f(x, y, α1, α2, ..., αn) = 0 ...(1)
where α1, α2, ..., αn are n independent parameters.
Equation (1) is known as an n parameter family of curves, e.g., y = mx is a one-parameter family of straight lines; x2 + y2 + ax + by = 0 is a two-parameters family of circles.
If we differentiate (1) n times w.r.t. x, we will get n more relations between x, y, α1, α2, ... αn and derivates of y w.r.t. x. By eliminating α1, α2, ..., αn from these n relations and (1), we get a differential equation.
Clearly the order of this differential equation will be n, i.e., equal to the number of independent parameters in the family of curves.
For example consider the family of parabolas with vertex at the origin and axis as the x-axis
y2 = 4ax ...(2)
Differentiating w.r.t. x, we get 2y 102245.png = 4a = 102239.png from (2) or, 2x 102233.pngy = 0, which is the differential equation of (2) and is clearly of order 1.

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