Differential Equations

First Order Equations

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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

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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.

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