Exercises In each exercise, obtain the differential equation of the family of plane curves described and sketch several representative members of the family. 1. Straight lines through the origin. Ans. y dx - x dy 0. hinoted
Exercises In each exercise, obtain the differential equation of the family of plane curves described and sketch several representative members of the family. 1. Straight lines through the origin. Ans. y dx - x dy 0. hinoted
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Kindly solve and show the solution on how to get the answer that is shown in the items no. 1,5,9,13 and 17. Thank you.
I hope you help me with these problems. I don't have much credits to ask another questions thank you.
![xy" – (y')³ – y' = 0.
Exercises
In each exercise, obtain the differential equation of the family of plane curves
described and sketch several representative members of the family.
Ans. y dx- x dy = 0.
1. Straight lines through the origin.
2. Straight lines through the fixed point (h, k); h and k not to be eliminated.
3. Straight lines with slope and y-intercept equal.
4, Straight lines with slope and x-intercept equal.
5. Straight lines with algebraic sum of the intercepts fixed as k.
Ans. (y- k) dx - (x h) dy = 0.
y dx - (x + 1) dy 0.
Ans. (y')? = xy - y.
Ans.
Ans. (xy'-yXy-1) + ky' = 0.
6. Straight lines at a fixed distance p from the origin.
7. Circles with center at the origin.
8. Circles with center on the x-axis.
9. Circles with fixed radius r and tangent to the x-axis.
Ans. (xy' y) = p²[1+ (y)].
Ans. x dx + y dy = 0.
Ans. yy" + (y)² + 1 = 0.
%3D
Ans. (y t r)(y')? + y? + 2ry 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4848a09c-86dc-44c2-8280-8974eb986c52%2F3a5535bf-1c63-4c68-a79b-8ae8daa3e0eb%2Fgue5rsp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:xy" – (y')³ – y' = 0.
Exercises
In each exercise, obtain the differential equation of the family of plane curves
described and sketch several representative members of the family.
Ans. y dx- x dy = 0.
1. Straight lines through the origin.
2. Straight lines through the fixed point (h, k); h and k not to be eliminated.
3. Straight lines with slope and y-intercept equal.
4, Straight lines with slope and x-intercept equal.
5. Straight lines with algebraic sum of the intercepts fixed as k.
Ans. (y- k) dx - (x h) dy = 0.
y dx - (x + 1) dy 0.
Ans. (y')? = xy - y.
Ans.
Ans. (xy'-yXy-1) + ky' = 0.
6. Straight lines at a fixed distance p from the origin.
7. Circles with center at the origin.
8. Circles with center on the x-axis.
9. Circles with fixed radius r and tangent to the x-axis.
Ans. (xy' y) = p²[1+ (y)].
Ans. x dx + y dy = 0.
Ans. yy" + (y)² + 1 = 0.
%3D
Ans. (y t r)(y')? + y? + 2ry 0.
![Sec. 4]
Families of Curves
15
Ans. (1 + (y)']' [yy"+i +(.
10. Circles tangent to the x-axis.
11. Circles with center on the line y -- X, and passing through the origin.
Ans. (x2- 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")? [I+ (y)'.
Ans. y"[1 + (y)'] 3y(y)?.
13. All circles. Use the curvature.
14. Paraboias with vertex on the x-axis, with axis parallel to the y-axis, and with
distance from focus to vertex fixed as a.
Ans. a(y)? = y.
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
Ans. 2ay" = 1.
fixed as a.
17. Parabolas with axis parallel to the x-axis and with distance from vertex to focus
Ans. 2ay" + (y) = 0.
d?x
* 1.
dy?
fixed as a.
Ans.
2a
18. Work Exercise 17, using differentiation with respect to y.
19. Use the fact that
"](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4848a09c-86dc-44c2-8280-8974eb986c52%2F3a5535bf-1c63-4c68-a79b-8ae8daa3e0eb%2Fur2xfkq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Sec. 4]
Families of Curves
15
Ans. (1 + (y)']' [yy"+i +(.
10. Circles tangent to the x-axis.
11. Circles with center on the line y -- X, and passing through the origin.
Ans. (x2- 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")? [I+ (y)'.
Ans. y"[1 + (y)'] 3y(y)?.
13. All circles. Use the curvature.
14. Paraboias with vertex on the x-axis, with axis parallel to the y-axis, and with
distance from focus to vertex fixed as a.
Ans. a(y)? = y.
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
Ans. 2ay" = 1.
fixed as a.
17. Parabolas with axis parallel to the x-axis and with distance from vertex to focus
Ans. 2ay" + (y) = 0.
d?x
* 1.
dy?
fixed as a.
Ans.
2a
18. Work Exercise 17, using differentiation with respect to y.
19. Use the fact that
"
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