# Differential Equations

1st Order Equations# Differential Equations of Plane Curves

Problems 1-2

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:

- Differentiate \(F\) with respect to \(x\) considering \(y\) as a function of \(x:\)

\[\frac{{\partial F}}{{\partial x}} + \frac{{\partial F}}{{\partial y}} \cdot y’ = 0;\] - 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.

Problems 1-2

Problems 3-5