# Differential Equations

1st Order Equations# Orthogonal Trajectories

Theory

Problems 1-4

### Definition and Examples

Let a family of curves be given by the equation

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 \(1\text{):}\)

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

Figure 1.

Figure 2.

Similarly, the orthogonal trajectories of the family of ellipses

are confocal hyperbolas satisfying the equation:

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:

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:

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:

- 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.
- 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.
- 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 \(y = C\sin x.\)

Theory

Problems 1-4