# Formation of Differential Equations

Consider a family of curves

*f*(

*x*,

*y*,

*Î±*

_{1},

*Î±*

_{2},

*...*,

*Î±*) = 0 ...(1)

_{n}where

*Î±*_{1},*Î±*_{2},*...*,*Î±*are_{n}*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;*x*^{2}+*y*^{2}+*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*,*Î±*_{1},*Î±*_{2},*...**Î±*and derivates of_{n}*y*w.r.t.*x*. By eliminating*Î±*_{1},*Î±*_{2},*...*,*Î±*from these_{n}*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*y*

^{2}= 4

*ax*...(2)

Differentiating w.r.t.

*x*, we get 2*y*= 4*a*= from (2) or, 2*x*â€“*y*= 0, which is the differential equation of (2) and is clearly of order 1.