Differential Equations - Theorem 1 on orthogonal trajectories and its examples
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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.
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Orthogonal trajectories are curves that intersect each curve of a given family at right angles.
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Let’s consider two families of curves: y = f(x, C1) and y = g(x, C2).
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C1 and C2 are constants that vary from curve to curve within their respective families.
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The slope of a curve y = f(x, C1) is given by: dy/dx = f’(x, C1).
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The slope of a curve y = g(x, C2) is given by: dy/dx = g’(x, C2).
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According to Theorem 1, we have the equation f’(x, C1) * g’(x, C2) = -1.
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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:
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Differentiate y = Cx with respect to x to find the slope of the family of curves.
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dy/dx = C
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Now, let’s substitute the slope (C) into the equation f’(x, C1) * g’(x, C2) = -1.
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C * g’(x, C2) = -1. We need to solve this equation to find the orthogonal trajectory.
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Solving the equation, we get: g’(x, C2) = -1/C.
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Integrating g’(x, C2) = -1/C with respect to x, we get g(x, C2) = -ln|C2| + K, where K is another constant.
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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:
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Differentiate y = Cx^n with respect to x to find the slope of the family of curves.
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dy/dx = nCx^(n-1)
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Substituting the slope into the equation f’(x, C1) * g’(x, C2) = -1, we get: nCx^(n-1) * g’(x, C2) = -1.
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Solving for g’(x, C2), we find g’(x, C2) = -1/(nCx^(n-1)).
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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.
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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:
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Differentiate y^2 = e^x with respect to x to find the slope of the family of curves.
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2yy’ = e^x
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Solving for y’, we get y’ = (e^x) / (2y).
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To find the orthogonal trajectory, we need to substitute the slope (y’) into the equation f’(x, C1) * g’(x, C2) = -1.
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(e^x) / (2y) * g’(x, C2) = -1
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Solving for g’(x, C2), we find g’(x, C2) = -2y / (e^x)
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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.
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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
- Orthogonal trajectories are curves that intersect each curve of a given family at right angles.
- Theorem 1 states that the product of the slopes of the two families of curves is -1 at any point of intersection.
- To find the orthogonal trajectories, differentiate the equation of the given family of curves with respect to x.
- Substitute the slope into the equation f’(x, C1) * g’(x, C2) = -1 and solve for the orthogonal trajectory.
- Examples help in understanding and applying the theorem to real problems.
Slide 16
- Further Practice
- Solve more examples involving different families of curves to strengthen your understanding.
- Explore the applications of orthogonal trajectories in physics and engineering.
- Attempt exercises from your textbook or other resources to enhance your problem-solving skills.
- Review other theorems and techniques in differential equations to expand your knowledge in this area.
- Seek help from your teacher or classmates if you face any difficulties or have questions.
Slide 17
- References
- Stewart, J., Redlin, L., & Watson, S. (2015). Precalculus: Mathematics for Calculus. Cengage Learning.
- Simmons, G. F., Galvan, F., Stead, G., & Kang, H. (2008). Differential Equations with Applications and Historical Notes. McGraw-Hill.
Slide 18
- Questions for Discussion
- Explain the concept of orthogonal trajectories and its significance in mathematics.
- How can the theorem on orthogonal trajectories help in solving differential equations?
- Give an example of two families of curves that are orthogonal trajectories of each other.
- In what real-world scenarios can the concept of orthogonal trajectories be applied?
- 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.
