The quartics of Exercise 26 with c held fixed and a to be eliminated.
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
Please answer item 27.
![15
Sec. 4]
Families of Curves
Ans. (1 + (y)']' = [yy"+1 + (y)²]?.
10. Circles tangent to the x-axis.
11. Circles with center on the line y = --x, and passing through the origin.
Ans. (x- 2xy - y?) dx + (x? + 2xy - y?) dy = 0.
12. Circles of radius unity. Use the fact that the radius of curvature is 1.
Ans. (y') = [1 + (y)²]³.
Ans. y"[1 + (y)²] = 3y(y")².
14. Parabolas with vertex on the x-axis, with axis parallel to the y-axis, and with
Ans. aly)? = y.
13. All circles, Use the curvature.
distance from focus to vertex fixed as a.
15. Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with
Ans. x(y)? = a.
distance from focus to vertex fixed as a.
16. Parabolas with axis parallel to the y-axis and with distance from vertex to focus
fixed as a.
Ans. 2ay" =1.
17. Parabolas with axis parallel to the x-axis and with distance from vertex to focus
fixed as a.
Ans. 2ay" + (}')³ = 0.
Ans.
2a
= 1.
18. Work Exercise 17, using differentiation with respect to y.
dy?
19. Use the fact that
--()-会点()
dx d (dx
dv dx dy/
d²x
d (dx
dx d
dy
%3D
dy?
dy (dy,
dy dx \dx,
(y)3
to prove that the answers to Exercises 17 and 18 are equivalent.
20. Parabolas with vertex and focus on the x-axis.
21. Parabolas with axis parallel to the x-axis.
22. Central conics with center at the origin and vertices on the coordinate axes.
Ans. yy" + (y)² = 0.
Ans. y'y" - 3(y")² = 0.
Aus. xyy" + x(y')² - yy = 0.
23. The confocal central cónics
x2
y2
+
= 1
a2 + A
b2 + i
Ans. (xy - yXyy + x) = (a² – b?)y'.
Ans. 2x(x - a)y' = y(3x - 2a).
with a and b held fixed.
24. The cubics cy? = x²(x – a) with a held fixed.
25. The cubics of Exercise 24 with c held fixed and a to be eliminated.
Ans. 2cy(xy- y) = x³.
Ans. 2x(x - a)y = y(4x – a).
26. The quartics c'y² = x(x - a)³ with a held fixed
27. The quartics of Exercise 26 with c held fixed and a to be eliminated.
Aus. c(2xy- y) = 27x y.
x²(a + x)
28. The strophoids y?
Ans. (x-4x²y2-y) dx + 4x'y dy = 0.
a - X
Ans. 2x'y = (y² + 3x)
29. The cissoids y?
a - x
30. The trisectrices of Maclaurin y'(a + x) = x²(3a - x).
Ans. (3x -6x'- +8y dy = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe105d5a6-9ca3-443d-b93b-6ff38da6f4df%2Fcaf98ea5-4f4f-4bed-a0fa-362f145416b5%2Fmdl4j38_processed.jpeg&w=3840&q=75)
Transcribed Image Text:15
Sec. 4]
Families of Curves
Ans. (1 + (y)']' = [yy"+1 + (y)²]?.
10. Circles tangent to the x-axis.
11. Circles with center on the line y = --x, and passing through the origin.
Ans. (x- 2xy - y?) dx + (x? + 2xy - y?) dy = 0.
12. Circles of radius unity. Use the fact that the radius of curvature is 1.
Ans. (y') = [1 + (y)²]³.
Ans. y"[1 + (y)²] = 3y(y")².
14. Parabolas with vertex on the x-axis, with axis parallel to the y-axis, and with
Ans. aly)? = y.
13. All circles, Use the curvature.
distance from focus to vertex fixed as a.
15. Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with
Ans. x(y)? = a.
distance from focus to vertex fixed as a.
16. Parabolas with axis parallel to the y-axis and with distance from vertex to focus
fixed as a.
Ans. 2ay" =1.
17. Parabolas with axis parallel to the x-axis and with distance from vertex to focus
fixed as a.
Ans. 2ay" + (}')³ = 0.
Ans.
2a
= 1.
18. Work Exercise 17, using differentiation with respect to y.
dy?
19. Use the fact that
--()-会点()
dx d (dx
dv dx dy/
d²x
d (dx
dx d
dy
%3D
dy?
dy (dy,
dy dx \dx,
(y)3
to prove that the answers to Exercises 17 and 18 are equivalent.
20. Parabolas with vertex and focus on the x-axis.
21. Parabolas with axis parallel to the x-axis.
22. Central conics with center at the origin and vertices on the coordinate axes.
Ans. yy" + (y)² = 0.
Ans. y'y" - 3(y")² = 0.
Aus. xyy" + x(y')² - yy = 0.
23. The confocal central cónics
x2
y2
+
= 1
a2 + A
b2 + i
Ans. (xy - yXyy + x) = (a² – b?)y'.
Ans. 2x(x - a)y' = y(3x - 2a).
with a and b held fixed.
24. The cubics cy? = x²(x – a) with a held fixed.
25. The cubics of Exercise 24 with c held fixed and a to be eliminated.
Ans. 2cy(xy- y) = x³.
Ans. 2x(x - a)y = y(4x – a).
26. The quartics c'y² = x(x - a)³ with a held fixed
27. The quartics of Exercise 26 with c held fixed and a to be eliminated.
Aus. c(2xy- y) = 27x y.
x²(a + x)
28. The strophoids y?
Ans. (x-4x²y2-y) dx + 4x'y dy = 0.
a - X
Ans. 2x'y = (y² + 3x)
29. The cissoids y?
a - x
30. The trisectrices of Maclaurin y'(a + x) = x²(3a - x).
Ans. (3x -6x'- +8y dy = 0
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