Differential Equations

1st Order Equations

Orthogonal Trajectories

Page 1
Theory
Page 2
Problems 1-4

Definition and Examples

Let a family of curves be given by the equation

\[g\left( {x,y} \right) = C,\]

where \(C\) is a constant. For the given family of curves, we can draw the orthogonal trajectories, i.e. another family of curves \(f\left( {x,y} \right) = C\) that cross the given curves at right angles.

For example, the orthogonal trajectory of the family of straight lines defined by the equation \(y = kx,\) where \(k\) is a parameter (the slope of the straight line), is any circle having centre at the origin (Figure \(1\text{):}\)

\[{x^2} + {y^2} = {R^2},\]

where \(R\) is the radius of the circle.

The orthogonal trajectory of the family of straight lines y=kx is any circle centered at the origin

Figure 1.

The orthogonal trajectories of the family of ellipses are confocal hyperbolas

Figure 2.

Similarly, the orthogonal trajectories of the family of ellipses

\[{\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{c^2} – {a^2}}} = 1,\;\;}\kern0pt{\text{where}\;\;}\kern-0.3pt{0 \lt a \lt c,}\]

are confocal hyperbolas satisfying the equation:

\[{\frac{{{x^2}}}{{{b^2}}} – \frac{{{y^2}}}{{{b^2} – {c^2}}} = 1,\;\;}\kern0pt{\text{where}\;\;}\kern-0.3pt{0 \lt c \lt b.}\]

Both families of curves are sketched in Figure \(2.\) Here \(a\) and \(b\) play the role of parameters describing the family of ellipses and hyperbolas, respectively.

General Method of Finding Orthogonal Trajectories

The common approach for determining orthogonal trajectories is based on solving the partial differential equation:

\[\nabla f\left( {x,y} \right) \cdot \nabla g\left( {x,y} \right) = 0,\]

where the symbol \(\nabla\) means the gradient of the function
\(f\left( {x,y} \right)\) or \(g\left( {x,y} \right)\) and the dot means the dot product of the two gradient vectors.

Using the definition of gradient, one can write:

\[{\nabla f\left( {x,y} \right) = \mathbf{grad}\,f\left( {x,y} \right) }={ \left( {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}}} \right),}\]
\[{\nabla g\left( {x,y} \right) = \mathbf{grad}\,g\left( {x,y} \right) }={ \left( {\frac{{\partial g}}{{\partial x}},\frac{{\partial g}}{{\partial y}}} \right).}\]

Hence, the partial differential equation is written in the form:

\[
{\nabla f\left( {x,y} \right) \cdot \nabla g\left( {x,y} \right) = 0,\;\;}\Rightarrow
{\left( {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}}} \right) \cdot \left( {\frac{{\partial g}}{{\partial x}},\frac{{\partial g}}{{\partial y}}} \right) = 0,\;\;}\Rightarrow
{\frac{{\partial f}}{{\partial x}}\frac{{\partial g}}{{\partial x}} + \frac{{\partial f}}{{\partial y}}\frac{{\partial g}}{{\partial y}} = 0.}
\]

Solving the last PDE, we can determine the equation of the orthogonal trajectories \(f\left( {x,y} \right) = C.\)

A Practical Algorithm for Constructing Orthogonal Trajectories

Below we describe an easier algorithm for finding orthogonal trajectories \(f\left( {x,y} \right) = C\) of the given family of curves \(g\left( {x,y} \right) = C\) using only ordinary differential equations. The algorithm includes the following steps:

  1. Construct the differential equation \(G\left( {x,y,y’} \right) = 0\) for the given family of curves \(g\left( {x,y} \right) = C.\) See the web page Differential Equations of Plane Curves about how to do this.
  2. Replace \(y’\) with \(\left( { – \large\frac{1}{{y’}}\normalsize} \right)\) in this differential equation. As a result, we obtain the differential equation of the orthogonal trajectories.
  3. Solve the new differential equation to determine the algebraic equation of the family of orthogonal trajectories \(f\left( {x,y} \right) = C.\)

Solved Problems

Click on problem description to see solution.

 Example 1

Find the orthogonal trajectories of the family of straight lines \(y = Cx,\) where \(C\) is a parameter.

 Example 2

A family of hyperbolic curves is given by the equation \(y = {\large\frac{C}{x}\normalsize}.\) Find the orthogonal trajectories for these curves.

 Example 3

Find the orthogonal trajectories of the family of curves given by the power function \(y = C{x^4}.\)

 Example 4

Determine the orthogonal trajectories of the family of sinusoids \(y = C\sin x.\)

Page 1
Theory
Page 2
Problems 1-4