# Find the ODE of a family of curves

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 A family of curves in the $xy$-plane is described by an equation of the form $y = f(x,y, C_1, ..., C_n)$ or $F(x, y, C_1, ..., C_n) = 0$. In order to find the ODE of such a family, one generally supplements its equation with $n$ additional equations generated through successive differentiations with respect to $x$. One ends up with $(n+1)$ equations from which the constants are eliminated to obtain the sought ODE. The following code implements this procedure geometricODE[eq_, indepVar_, depVar_, cst_] := Eliminate[ D[eq /. a_ == b_ :> a - b /. depVar -> depVar@indepVar, {indepVar, #}] == 0 & /@ Range[0, Length[cst]], cst] // Simplify; Below are few examples. geometricODE[x^2*y^2 - b^2*x^2 - a^2*y^2, x, y, {a, b}] geometricODE[(x - a)^2 + (y - b)^2 == c^2, x, y, {a, b, c}] Solve[geometricODE[y == Tan[a*x + b] + 1/2, x, y, {a, b}], y''[x]] // Quiet Solve[geometricODE[y - c*Sqrt[x]* Cos[a*Sqrt[x] + b] == 0, x, y, {a, b, c}], y'''[x]] // Quiet Answer