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

    Click 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 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