Differential Equations of Plane Curves

As it is known, the solution of a differential equation is displayed graphically as a family of integral curves. It turns out that one can also solve the inverse problem: construct a differential equation of the family of plane curves defined by an algebraic equation!

Suppose that a family of plane curves is described by the implicit one-parameter equation:

We assume that the function \(F\) has continuous partial derivatives in \(x\) and \(y.\) To write the corresponding differential equation of first order, it's necessary to perform the following steps:

1. Differentiate \(F\) with respect to \(x\) considering \(y\) as a function of \(x:\)
   \[\frac{{\partial F}}{{\partial x}} + \frac{{\partial F}}{{\partial y}} \cdot y' = 0;\]
2. Solve the system of equations:
   \[\left\{ \begin{array}{l} \frac{{\partial F}}{{\partial x}} + \frac{{\partial F}}{{\partial y}} \cdot y' = 0\\ F\left( {x,y,C} \right) = 0 \end{array} \right.\]
   by eliminating the parameter \(C\) from it.

If a family of plane curves is given by the two-parameter equation

we should differentiate the last formula twice by considering \(y\) as a function of \(x\) and then eliminating the parameters \({{C_1}}\) and \({{C_2}}\) from the system of three equations.

The similar rule is applied to the case of \(n\)-parametric family of plane curves.

Solved Problems

Solution.

Differentiating the given equation with respect to \(x\) gives:

We can easily eliminate the parameter \(C\) from the system of equations:

As a result, we obtain the following simplest homogeneous equation:

Solution.

We differentiate the implicit equation with respect to \(x:\)

Write this equation jointly with the original algebraic equation and eliminate the parameter \(C:\)

As a result, we obtain the implicit differential equation corresponding to the given family of plane curves.