# Differential Equations

## First Order Equations # 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:

$F\left( {x,y,C} \right) = 0.$

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

$F\left( {x,y,{C_1},{C_2}} \right) = 0,$

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

Click or tap a problem to see the solution.

### Example 1

Determine the differential equation for the family of curves defined by the equation $y = {e^{x + C}}.$

### Example 2

Derive the differential equation for the family of plane curves defined by the equation $y = {x^2} - Cx.$

### Example 1.

Determine the differential equation for the family of curves defined by the equation $y = {e^{x + C}}.$

Solution.

Differentiating the given equation with respect to $$x$$ gives:

$y' = {e^{x + C}}.$

We can easily eliminate the parameter $$C$$ from the system of equations:

$\left\{ \begin{array}{l} y' = {e^{x + C}}\\ y = {e^{x + C}} \end{array} \right..$

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

$y' = y,\;\; \Rightarrow y' - y = 0.$

### Example 2.

Derive the differential equation for the family of plane curves defined by the equation $y = {x^2} - Cx.$

Solution.

We differentiate the implicit equation with respect to $$x:$$

$y' = 2x - C.$

Write this equation jointly with the original algebraic equation and eliminate the parameter $$C:$$

$\left\{ \begin{array}{l} y' = 2x - C\\ y = {x^2} - Cx \end{array} \right.,\;\; \Rightarrow C = 2x - y',\;\; \Rightarrow y = {x^2} - \left( {2x - y'} \right)x,\;\; \Rightarrow y = {x^2} + y'x - 2{x^2},\;\; \Rightarrow y'x - y = {x^2}.$

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

See more problems on Page 2.