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# Differential Equations

1st Order Equations

# Differential Equations of Plane Curves

Page 1
Problems 1-2
Page 2
Problems 3-5

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 on problem description to see 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 3

Write the corresponding differential equation for the family of plane curves defined by the equation $$y = \cot \left( {x – C} \right).$$

### ✓Example 4

A family of curves is given by the expression $$y =$$ $${\large\frac{1}{C}\normalsize}\cos \left( {Cx + \alpha } \right),$$ where $$C$$ is a parameter, $$\alpha$$ is an arbitrary angle. Determine the differential equation for this family of plane curves.

### ✓Example 5

Derive the differential equation for the family of two-parameter plane curves $$y = {C_1}{x^2} + {C_2}x.$$

### 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 $$C:$$

${\left\{ \begin{array}{l} y’ = 2x – C\\ y = {x^2} – Cx \end{array} \right.,\;\;}\Rightarrow {C = y’ – 2x,\;\;}\Rightarrow {y = {x^2} – \left( {y’ – 2x} \right)x,\;\;}\Rightarrow {y = {x^2} – y’x + 2{x^2},\;\;}\Rightarrow {y’x + y = 3{x^2}.}$

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

Page 1
Problems 1-2
Page 2
Problems 3-5