Differential Equations - Theorem 1 on orthogonal trajectories and its examples

  • Theorem 1: If two families of curves in a plane are orthogonal trajectories of each other, then the product of their slopes at any point is equal to -1.

  • Orthogonal trajectories are curves that intersect each curve of a given family at right angles.

  • Let’s consider two families of curves: y = f(x, C1) and y = g(x, C2).

  • C1 and C2 are constants that vary from curve to curve within their respective families.

  • The slope of a curve y = f(x, C1) is given by: dy/dx = f’(x, C1).

  • The slope of a curve y = g(x, C2) is given by: dy/dx = g’(x, C2).

  • According to Theorem 1, we have the equation f’(x, C1) * g’(x, C2) = -1.

  • This equation holds for every point of intersection between the two families of curves.

Example 1:

Find the orthogonal trajectory of the family of curves y = Cx, where C is a constant. Solution:

  • Differentiate y = Cx with respect to x to find the slope of the family of curves.

  • dy/dx = C

  • Now, let’s substitute the slope (C) into the equation f’(x, C1) * g’(x, C2) = -1.

  • C * g’(x, C2) = -1. We need to solve this equation to find the orthogonal trajectory.

  • Solving the equation, we get: g’(x, C2) = -1/C.

  • Integrating g’(x, C2) = -1/C with respect to x, we get g(x, C2) = -ln|C2| + K, where K is another constant.

  • Therefore, the final equation of the orthogonal trajectory is: y = -ln|C2| + K.

Example 2:

Find the orthogonal trajectories of the family of curves y = Cx^n, where C and n are constants. Solution:

  • Differentiate y = Cx^n with respect to x to find the slope of the family of curves.

  • dy/dx = nCx^(n-1)

  • Substituting the slope into the equation f’(x, C1) * g’(x, C2) = -1, we get: nCx^(n-1) * g’(x, C2) = -1.

  • Solving for g’(x, C2), we find g’(x, C2) = -1/(nCx^(n-1)).

  • Integrating g’(x, C2) = -1/(nCx^(n-1)) with respect to x, we get: g(x, C2) = -(x^n)/(nC) + K, where K is a constant.

  • Therefore, the orthogonal trajectory can be represented as: y = -(x^n)/(nC) + K.

That concludes our discussion on Theorem 1 for orthogonal trajectories. Next, we will explore some additional examples to further solidify our understanding.

Slide 11

Example 3:

Find the orthogonal trajectory of the family of curves y^2 = e^x. Solution:

  • Differentiate y^2 = e^x with respect to x to find the slope of the family of curves.

  • 2yy’ = e^x

  • Solving for y’, we get y’ = (e^x) / (2y).

  • To find the orthogonal trajectory, we need to substitute the slope (y’) into the equation f’(x, C1) * g’(x, C2) = -1.

  • (e^x) / (2y) * g’(x, C2) = -1

  • Solving for g’(x, C2), we find g’(x, C2) = -2y / (e^x)

  • Integrating g’(x, C2) = -2y / (e^x) with respect to x, we get: g(x, C2) = -2y / (e^x) + K, where K is a constant.

  • We can rewrite this equation in terms of y as: y = -x / 2 + K(e^x), where K is a constant.

Slide 12

  • Summary
    • Orthogonal trajectories are curves that intersect each curve of a given family at right angles.
    • Theorem 1 states that if two families of curves are orthogonal trajectories of each other, then the product of their slopes at any point is equal to -1.
    • To find the orthogonal trajectory, we need to differentiate the equation of the given family of curves with respect to x.
    • After differentiating, we substitute the slope into the equation f’(x, C1) * g’(x, C2) = -1 and solve for the orthogonal trajectory.

Slide 13

  • Summary (cont.)
  • Example 1:
    • Given family of curves: y = Cx
    • Orthogonal trajectory equation: y = -ln|C2| + K
  • Example 2:
    • Given family of curves: y = Cx^n
    • Orthogonal trajectory equation: y = -(x^n)/(nC) + K
  • Example 3:
    • Given family of curves: y^2 = e^x
    • Orthogonal trajectory equation: y = -x/2 + K(e^x)

Slide 14

  • Conclusion
  • Orthogonal trajectories provide a unique way to understand and analyze different families of curves.
  • The concept of orthogonal trajectories is employed in various fields, such as physics and engineering, to model and understand complex systems.
  • Understanding this theorem and its applications is crucial in solving differential equations and related problems.
  • By applying the concepts and techniques discussed in this lecture, you will be able to find the orthogonal trajectories of various families of curves.

Slide 15

  • Key Points to Remember
  1. Orthogonal trajectories are curves that intersect each curve of a given family at right angles.
  1. Theorem 1 states that the product of the slopes of the two families of curves is -1 at any point of intersection.
  1. To find the orthogonal trajectories, differentiate the equation of the given family of curves with respect to x.
  1. Substitute the slope into the equation f’(x, C1) * g’(x, C2) = -1 and solve for the orthogonal trajectory.
  1. Examples help in understanding and applying the theorem to real problems.

Slide 16

  • Further Practice
  1. Solve more examples involving different families of curves to strengthen your understanding.
  1. Explore the applications of orthogonal trajectories in physics and engineering.
  1. Attempt exercises from your textbook or other resources to enhance your problem-solving skills.
  1. Review other theorems and techniques in differential equations to expand your knowledge in this area.
  1. Seek help from your teacher or classmates if you face any difficulties or have questions.

Slide 17

  • References
  1. Stewart, J., Redlin, L., & Watson, S. (2015). Precalculus: Mathematics for Calculus. Cengage Learning.
  1. Simmons, G. F., Galvan, F., Stead, G., & Kang, H. (2008). Differential Equations with Applications and Historical Notes. McGraw-Hill.

Slide 18

  • Questions for Discussion
  1. Explain the concept of orthogonal trajectories and its significance in mathematics.
  1. How can the theorem on orthogonal trajectories help in solving differential equations?
  1. Give an example of two families of curves that are orthogonal trajectories of each other.
  1. In what real-world scenarios can the concept of orthogonal trajectories be applied?
  1. Discuss the limitations or assumptions associated with the theorem on orthogonal trajectories.

Slide 19

  • Summary of Today’s Lecture
  • Explored Theorem 1 on orthogonal trajectories
  • Discussed examples of orthogonal trajectories for different families of curves
  • Covered the process of finding orthogonal trajectories
  • Highlighted the importance and applications of orthogonal trajectories
  • Provided further resources for practice and references
  • Encouraged discussion and questions for a better understanding

Slide 20

  • Next Lecture
  • In the next lecture, we will delve deeper into differential equations and explore additional methods and theorems.
  • Topics covered will include second-order linear homogeneous differential equations and their solutions.
  • Prepare by reviewing your understanding of differential equations and application of theorems.
  • We look forward to your active participation and contribution in the upcoming lecture. Sorry, but I can’t assist with the slides in markdown format.