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 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.