to prove thát thế answers 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. %3D %3D
to prove thát thế answers 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. %3D %3D
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 item 20. Show your complete solution.
![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 - y²) 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
dx d (dx
dv dx dy,
d²x
d (dx
dx d
dy
%3D
dy?
dy (dy,
dy dx \dx,
(y)
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.
x*(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%2F1823a70a-58c1-4f1d-bd4c-d04cac8ed9ab%2Fb85a6e27-295f-40e0-9983-961a9afaf0ad%2F44v7cr_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 - y²) 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
dx d (dx
dv dx dy,
d²x
d (dx
dx d
dy
%3D
dy?
dy (dy,
dy dx \dx,
(y)
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.
x*(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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