Consider the case when S[y] = [" √1+32 dx y(0) = 8, 0 y where y(v) > 0, 5 > 0 and the right-hand end point (v, y(v)) lies on the line ay +ẞx+y= 0, where a, ß, y are constants with ẞ +0. Show that the first-integral may be written as y√1+y = C for some constant c > 0, and that the solutions of the first-integral are circles centred at (-cs, 0) with radius C₁ i.e. y² + (x + cs)² = c², where c² = c² - 8². * Using the transversality condition and by differentiating implicitly the equation y²+(x + cs)² = c², show that y' = a/ẞ and cs = y/ß. Finally, show that in the limit as → 0, the stationary path becomes ß²y² + (ßx + y)² = 7².
Consider the case when S[y] = [" √1+32 dx y(0) = 8, 0 y where y(v) > 0, 5 > 0 and the right-hand end point (v, y(v)) lies on the line ay +ẞx+y= 0, where a, ß, y are constants with ẞ +0. Show that the first-integral may be written as y√1+y = C for some constant c > 0, and that the solutions of the first-integral are circles centred at (-cs, 0) with radius C₁ i.e. y² + (x + cs)² = c², where c² = c² - 8². * Using the transversality condition and by differentiating implicitly the equation y²+(x + cs)² = c², show that y' = a/ẞ and cs = y/ß. Finally, show that in the limit as → 0, the stationary path becomes ß²y² + (ßx + y)² = 7².
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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![Consider the case when
S[y] = ["
√1+32
dx
y(0) = 8,
0
y
where y(v) > 0, 5 > 0 and the right-hand end point (v, y(v)) lies on the
line ay +ẞx+y= 0, where a, ß, y are constants with ẞ +0.
Show that the first-integral may be written as
y√1+y
= C
for some constant c > 0, and that the solutions of the first-integral are
circles centred at (-cs, 0) with radius C₁ i.e.
y² + (x + cs)² = c²,
where c² = c² - 8².
* Using the transversality condition and by differentiating implicitly the
equation y²+(x + cs)² = c², show that y' = a/ẞ and cs = y/ß.
Finally, show that in the limit as → 0, the stationary path becomes
ß²y² + (ßx + y)² = 7².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2F93a0b2df-ae65-43d2-9e5c-a3c7a20bb738%2Fefpjs0a_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the case when
S[y] = ["
√1+32
dx
y(0) = 8,
0
y
where y(v) > 0, 5 > 0 and the right-hand end point (v, y(v)) lies on the
line ay +ẞx+y= 0, where a, ß, y are constants with ẞ +0.
Show that the first-integral may be written as
y√1+y
= C
for some constant c > 0, and that the solutions of the first-integral are
circles centred at (-cs, 0) with radius C₁ i.e.
y² + (x + cs)² = c²,
where c² = c² - 8².
* Using the transversality condition and by differentiating implicitly the
equation y²+(x + cs)² = c², show that y' = a/ẞ and cs = y/ß.
Finally, show that in the limit as → 0, the stationary path becomes
ß²y² + (ßx + y)² = 7².
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