Differential Equations - Differential equation of the orthogonal trajectories

• In this lecture, we will discuss the concept of orthogonal trajectories in the context of differential equations.
• We will learn how to find the differential equation of the orthogonal trajectories.
• Orthogonal trajectories are a set of curves that intersect each member of another set of curves at right angles.

Orthogonal trajectories - Intuition

• To understand orthogonal trajectories, consider two families of curves given by the equations:
  • Equation 1: F(x, y, c1) = 0, where c1 is a parameter
  • Equation 2: G(x, y, c2) = 0, where c2 is a parameter
• The curves in Equation 1 intersect the curves in Equation 2 at right angles.
• The differential equation that represents the orthogonal trajectories can be derived using calculus.

Finding the differential equation of orthogonal trajectories - Step 1

1. Start by differentiating Equation 1 with respect to x, treating y as the dependent variable.

2. We obtain the derivative dy/dx in terms of x, y, and c1.

Finding the differential equation of orthogonal trajectories - Step 2

3. Similarly, differentiate Equation 2 with respect to x, treating y as the dependent variable.

4. This time, we obtain the derivative dy/dx in terms of x, y, and c2.

Finding the differential equation of orthogonal trajectories - Step 3

5. Take the reciprocal of the derivative obtained in Step 2.

6. This is done to ensure that the two derivatives are reciprocals of each other, representing perpendicular lines.

Finding the differential equation of orthogonal trajectories - Step 4

7. Equate the two reciprocals of the derivatives obtained in Steps 4 and 5.

8. This equation will give us the desired differential equation.

Example

Consider the family of curves given by the equation x^2 - y^2 = c.

• Differentiating with respect to x, we get: 2x - 2yy’ = 0.
• Differentiating with respect to x again, we get: 2 - 2y(y’)^2 - 2yy’’ = 0.
• Solving these two equations, we obtain the differential equation for the orthogonal trajectories: y’’ - y / x = 0.

Properties of orthogonal trajectories

• Orthogonal trajectories have the property that at any point of intersection, the tangents are perpendicular.
• The angle between the tangent to one curve and the tangent to its orthogonal trajectory is always 90 degrees.
• Orthogonal trajectories can exist for certain families of curves, while others may not have orthogonal trajectories.

Equations that do not have orthogonal trajectories

• Some equations do not possess orthogonal trajectories. Examples include:
  • Linear equations of the form ax + by + c = 0
  • Homogeneous equations of the form F(ax+by) = 0
  • Equations with exponential terms of the form y = Ke^ax

Summary

• Orthogonal trajectories are curves that intersect each member of another set of curves at right angles.
• The differential equation of orthogonal trajectories can be obtained by equating the derivatives of the curves.
• Orthogonal trajectories have the property that their tangents are perpendicular at points of intersection.
• Certain equations may not have orthogonal trajectories, such as linear equations and equations with exponential terms.

Differential Equations - Differential equation of the orthogonal trajectories

• Orthogonal trajectories are a set of curves that intersect each member of another set of curves at right angles.

• The differential equation that represents the orthogonal trajectories can be derived using calculus.

• To find the differential equation of orthogonal trajectories, follow these steps:

Step 1: Differentiate the given equation

• Start by differentiating the equation representing the family of curves with respect to x, treating y as the dependent variable.

• Express the derivative dy/dx in terms of x, y, and a parameter, say c1.

• Example:

  • Consider the family of curves given by the equation y = mx + c.
  • Differentiating with respect to x, we get: dy/dx = m.

Step 2: Differentiate the equation again

• Next, differentiate the equation obtained in Step 1 with respect to x, treating y as the dependent variable.

• Express the derivative dy/dx in terms of x, y, and a new parameter, say c2.

• Example:

  • Using the equation obtained in Step 1, dy/dx = m.
  • Differentiating with respect to x again, we get: d^2y/dx^2 = 0.

Step 3: Take the reciprocal

• Take the reciprocal of the derivative obtained in Step 2.

• This is done to ensure that the two derivatives are reciprocals of each other, representing perpendicular lines.

• Example:

  • Using the equation obtained in Step 2, d^2y/dx^2 = 0.
  • Taking the reciprocal, we get: 1/(d^2y/dx^2) = undefined (as the derivative is zero).

Step 4: Equate the reciprocals

• Equate the two reciprocals of the derivatives obtained in Steps 2 and 3.

• This equation will give us the desired differential equation of the orthogonal trajectories.

• Example:

  • Equating the reciprocals, we get: m = undefined.

Example

Consider the family of curves given by the equation y^2 = 4ax.

• Differentiating with respect to x, we get: 2yy’ = 4a.

• Differentiating with respect to x again, we get: 2(y’)^2 + 2yy’’ = 0.

• Taking the reciprocal, we get: 1/(2(y’)^2 + 2yy’’) = undefined.

• Equating the reciprocals, we get: 4(x/a) = undefined.

Properties of orthogonal trajectories

• Orthogonal trajectories have the property that at any point of intersection, the tangents are perpendicular.

• The angle between the tangent to one curve and the tangent to its orthogonal trajectory is always 90 degrees.

• Orthogonal trajectories can exist for certain families of curves, while others may not have orthogonal trajectories.

• Example:

  • In the case of straight lines, the orthogonal trajectories are also straight lines.
  • In the case of circles, the orthogonal trajectories are also circles.

Equations that do not have orthogonal trajectories

• Some equations do not possess orthogonal trajectories. Examples include:

  • Linear equations of the form ax + by + c = 0
  • Homogeneous equations of the form F(ax+by) = 0
  • Equations with exponential terms of the form y = Ke^ax

• Example:

  • The equation of a circle x^2 + y^2 = r^2 does not have orthogonal trajectories.

Summary

• Orthogonal trajectories are curves that intersect each member of another set of curves at right angles.

• The differential equation of orthogonal trajectories can be obtained by equating the derivatives of the curves.

• Orthogonal trajectories have the property that their tangents are perpendicular at points of intersection.

• Certain equations may not have orthogonal trajectories, such as linear equations and equations with exponential terms.

• Example:

  • The differential equation of orthogonal trajectories for the family of curves y = mx + c is m = undefined. These slides contain the rest of the content requested:

Identifying Orthogonal Trajectories

• Orthogonal trajectories can be found by solving the differential equation derived in the previous steps.

• The solutions to this differential equation will represent the orthogonal trajectories to the given family of curves.

• Example:

  • For the differential equation y’’ - y / x = 0, the solutions represent the orthogonal trajectories to the family of curves given by x^2 - y^2 = c.

Solving Differential Equations

• The process of finding the solution to a differential equation involves:

  • Classifying the equation as either linear or nonlinear
  • Identifying the order of the equation (whether it is first-order or higher-order)
  • Choosing an appropriate method to solve the equation

• Example:

  • For the differential equation y’’ - y / x = 0, we can use the method of separation of variables to find the solution.

Geometric Interpretation

• The geometric interpretation of orthogonal trajectories can be understood by considering the family of curves and their orthogonal counterparts.

• By analyzing the slopes of the curves, their intersections, and the angles between their tangents, one can observe the orthogonal nature of the trajectories.

• Example:

  • For the family of curves y = mx + c, the orthogonal trajectories are perpendicular to each curve at the point of intersection.

Applications of Orthogonal Trajectories

• Orthogonal trajectories have various applications in fields such as physics, engineering, and computer graphics.

• They can be used to model physical phenomena, design antennas, optimize thermal insulation, and create realistic shading in computer graphics.

• Example:

  • In physics, the equipotential lines in an electric field are orthogonal trajectories to the lines of force.

Understanding Orthogonality in Curves

• Two curves are said to be orthogonal if their tangents at the point of intersection are perpendicular to each other.

• The concept of orthogonality can be extended to surfaces in three-dimensional space, where tangent planes are considered instead of tangents.

• Example:

  • The curves y = mx + c and y = -1/mx + c’ are orthogonal to each other.

Orthogonal Trajectories in Polar Coordinates

• Orthogonal trajectories can also be determined in polar coordinates.

• In this case, the differential equation representing the orthogonal trajectories is obtained by differentiating the equation representing the family of curves with respect to the angle instead of x.

• Example:

  • For the family of curves r = a cosθ, the differential equation of the orthogonal trajectories is given by (θ’)^2 - cotθ = 0.

Finding Orthogonal Trajectories Using Symmetry

• In some cases, the presence of symmetry in the given family of curves can help in determining the orthogonal trajectories.

• By considering the properties of the given curves, one can deduce the properties of their orthogonal counterparts.

• Example:

  • For a set of concentric circles, the orthogonal trajectories are lines radiating from the center.

Orthogonal Trajectories in Vector Fields

• Orthogonal trajectories can also be defined in vector fields.

• In this case, the orthogonal trajectory represents the path taken by a particle moving in the direction perpendicular to the vector field.

• Example:

  • In the case of a magnetic field, the orthogonal trajectory represents the path followed by a charged particle moving perpendicular to the magnetic field lines.

Summary

• Orthogonal trajectories are curves that intersect each member of another set of curves at right angles.

• The differential equation of orthogonal trajectories can be derived by equating the derivatives of the curves.

• Solutions to this differential equation represent the orthogonal trajectories.

• Orthogonal trajectories have various applications in different fields, including physics and engineering.

• Example:

  • The differential equation of orthogonal trajectories for the family of curves x^2 - y^2 = c is given by y’’ - y / x = 0.

Review Questions

1. What are orthogonal trajectories?

2. How can the differential equation of orthogonal trajectories be derived?

3. What are some properties of orthogonal trajectories?

4. Which equations do not have orthogonal trajectories?

5. Give an example of an application of orthogonal trajectories.

• Example:
  • Solve the following differential equation to find the orthogonal trajectories: y’’ + y/x = 0.