20. Parabolas with vertex and focus on the x-axis.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Kindly answer items no. 20 and 27
![## Families of Curves
### Exercises:
10. **Circles tangent to the x-axis.**
- **Answer:** \([1 + (y')^2]^3 = [y'' y + 1 + (y')^2]^2\).
11. **Circles with center on the line \( y = -x \), and passing through the origin.**
- **Answer:** \((x^2 - 2xy - y^2) dx + (x^2 - 2xy - y^2) dy = 0\).
12. **Circles of radius unity. Use the fact that the radius of curvature is 1.**
- **Answer:** \( (y')^2 = [1 + (y')^2]^3 \).
13. **All circles. Use the curvature.**
- **Answer:** \( y''[1 + (y')^2] = 3(y')y''\).
14. **Parabolas with vertex on the x-axis, with axis parallel to the y-axis, and with distance from focus to vertex fixed as \( a \).**
- **Answer:** \( a(y')^2 = y \).
15. **Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from focus to vertex fixed as \( a \).**
- **Answer:** \( x(y')^2 = a \).
16. **Parabolas with axis parallel to the y-axis and with distance from vertex to focus fixed as \( a \).**
- **Answer:** \( 2ay = 1 \).
17. **Parabolas with axis parallel to the x-axis and with distance from vertex to focus fixed as \( a \).**
- **Answer:** \( 2ay + (y')^3 = 0 \).
18. **Work Exercise 17, using differentiation with respect to \( y \).**
- **Answer:** \( 2a \frac{d^2x}{dy^2} = 1 \).
19. **Use the fact that:**
\[
\frac{d^2x}{dy^2} \left( \frac{d}{dy} \left( \frac{dx}{dy} \right) \right)
= \frac{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe105d5a6-9ca3-443d-b93b-6ff38da6f4df%2F91ee6770-10d2-433a-b594-45ffe7996e65%2Foy0jsuf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Families of Curves
### Exercises:
10. **Circles tangent to the x-axis.**
- **Answer:** \([1 + (y')^2]^3 = [y'' y + 1 + (y')^2]^2\).
11. **Circles with center on the line \( y = -x \), and passing through the origin.**
- **Answer:** \((x^2 - 2xy - y^2) dx + (x^2 - 2xy - y^2) dy = 0\).
12. **Circles of radius unity. Use the fact that the radius of curvature is 1.**
- **Answer:** \( (y')^2 = [1 + (y')^2]^3 \).
13. **All circles. Use the curvature.**
- **Answer:** \( y''[1 + (y')^2] = 3(y')y''\).
14. **Parabolas with vertex on the x-axis, with axis parallel to the y-axis, and with distance from focus to vertex fixed as \( a \).**
- **Answer:** \( a(y')^2 = y \).
15. **Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from focus to vertex fixed as \( a \).**
- **Answer:** \( x(y')^2 = a \).
16. **Parabolas with axis parallel to the y-axis and with distance from vertex to focus fixed as \( a \).**
- **Answer:** \( 2ay = 1 \).
17. **Parabolas with axis parallel to the x-axis and with distance from vertex to focus fixed as \( a \).**
- **Answer:** \( 2ay + (y')^3 = 0 \).
18. **Work Exercise 17, using differentiation with respect to \( y \).**
- **Answer:** \( 2a \frac{d^2x}{dy^2} = 1 \).
19. **Use the fact that:**
\[
\frac{d^2x}{dy^2} \left( \frac{d}{dy} \left( \frac{dx}{dy} \right) \right)
= \frac{
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
