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.
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.
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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WITH BRIEF EXPLANATION IF POSSIBLE. THANK YOU.
![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.
2. Straight lines through the fixed point (h, k); h and k not to be eliminated.
ANS. y dx x dy = 0.
ANS. (y k) dx − (x − h) dy = 0.
-
ANS.
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.
7. Circles with center at the origin.
8. Circles with center on the x-axis.
y dx
9. Circles with fixed radius r and tangent to the x-axis.
ANS.
10. Circles tangent to the x-axis.
ANS.
(x + 1) dy = 0.
-
ANS. (xy' — y)(y' − 1) + ky' = 0.
6. Straight lines at a fixed distance p from the origin.
ANS.
(y)² = xy' — y.
-
(xy' - y)² = p²[1 + (y')²].
ANS. x dx + y dy = 0.
ANS. yy" + (y)² + 1 = 0.
(y ± r)²(y')² + y² ± 2ry = 0.
ANS. [1 + (y')²]³ = [yy″ + 1 + (y')²]².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8943d76e-c254-4322-8f39-c91cdf9301f3%2Fdc20a6bd-1957-41a0-a9f6-83be38019c69%2Fhxqb80b_processed.png&w=3840&q=75)
Transcribed Image Text: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.
2. Straight lines through the fixed point (h, k); h and k not to be eliminated.
ANS. y dx x dy = 0.
ANS. (y k) dx − (x − h) dy = 0.
-
ANS.
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.
7. Circles with center at the origin.
8. Circles with center on the x-axis.
y dx
9. Circles with fixed radius r and tangent to the x-axis.
ANS.
10. Circles tangent to the x-axis.
ANS.
(x + 1) dy = 0.
-
ANS. (xy' — y)(y' − 1) + ky' = 0.
6. Straight lines at a fixed distance p from the origin.
ANS.
(y)² = xy' — y.
-
(xy' - y)² = p²[1 + (y')²].
ANS. x dx + y dy = 0.
ANS. yy" + (y)² + 1 = 0.
(y ± r)²(y')² + y² ± 2ry = 0.
ANS. [1 + (y')²]³ = [yy″ + 1 + (y')²]².
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