arc of considerab. mterest tO cquation has the form ns. A Riccati dy = A(r)y + B(x)y + C(r). dr The mcthod of solution was discovercd by Eulcr. a) Show that if y1 (x) is a solution to the Riccati cquation then the substitution y= + transforms 1. the cquation into the first order lincar cquation dv + (B + 2AY1)v = -A. dx b) Use the method of part (a) to find the gencral solution for the Riccati cquation y? - 2ry + 1 +1 dx given that y(r) = z is a solution to this cquation. (Answer: y(1) = 1+ (c-r)) The following application of Riccati equations is due to Z. Hearon (1952). The propagation of a single action in a large population (drivers turning on hcadlights for example) often depends partly on cxternal circumstances (gathering darkncss) and partly on a tendency to imitate others who have alrcady performed the action. In this case, Hearon argucd that the proportion y(t) who have performed the action can be described by the cquation y (t) = (1– y)[g(t) + by). where g(t) measurcs the cxtcrnal stimulus and b is the imitation cocfficient. c) Notc that yı (t) = 1 is a solution to this Riccati cquation. Usc your rcsult from part (a) to find a lincar equation satisficd by v. (Answer: v+ (-6-g(t))v b)) d) Solve the differential cquation that you derived in part (c) for the function v in the special case g(t) = at. Lcave your solution in terms of an integral. (Answer: v(t) = e(bl+)S bel-bt-dt + ce+)
arc of considerab. mterest tO cquation has the form ns. A Riccati dy = A(r)y + B(x)y + C(r). dr The mcthod of solution was discovercd by Eulcr. a) Show that if y1 (x) is a solution to the Riccati cquation then the substitution y= + transforms 1. the cquation into the first order lincar cquation dv + (B + 2AY1)v = -A. dx b) Use the method of part (a) to find the gencral solution for the Riccati cquation y? - 2ry + 1 +1 dx given that y(r) = z is a solution to this cquation. (Answer: y(1) = 1+ (c-r)) The following application of Riccati equations is due to Z. Hearon (1952). The propagation of a single action in a large population (drivers turning on hcadlights for example) often depends partly on cxternal circumstances (gathering darkncss) and partly on a tendency to imitate others who have alrcady performed the action. In this case, Hearon argucd that the proportion y(t) who have performed the action can be described by the cquation y (t) = (1– y)[g(t) + by). where g(t) measurcs the cxtcrnal stimulus and b is the imitation cocfficient. c) Notc that yı (t) = 1 is a solution to this Riccati cquation. Usc your rcsult from part (a) to find a lincar equation satisficd by v. (Answer: v+ (-6-g(t))v b)) d) Solve the differential cquation that you derived in part (c) for the function v in the special case g(t) = at. Lcave your solution in terms of an integral. (Answer: v(t) = e(bl+)S bel-bt-dt + ce+)
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
100%
![arc of considerab.c nterest tom
cquation has the form
ns. A Riccati
dy
= A(r)y+ B(x)y + C(x).
dr
The mcthod of solution was discovercd by Eulcr.
a) Show that if y1 (x) is a solution to the Riccati cquation then the substitution y= y+ transforms
1.
the cquation into the first order lincar cquation
dv
+ (B+2AY1)v = -A.
dx
b) Use the method of part (a) to find the general solution for the Riccati cquation
dy
y? - 2ry + 1 + 1
dx
given that y(r)
= z is a solution to this cquation. (Answer: y(1) = 1+ (c-r))
The following application of Riccati cquations is due to Z. Hearon (1952). The propagation of a
single action in a large population (drivers turning on hcadlights for example) often depends partly
on cxtcrnal circumstances (gathering darkncss) and partly on a tendency to imitate others who
have alrcady performed the action. In this case, Hearon argucd that the proportion y(t) who have
performed the action can be described by the cquation
y (t) = (1– y)[g(t) + by].
whcre g(t) mcasurcs the cxtcrnal stimulus and b is the imitation cocfficient.
c) Notc that yı (t) = 1 is a solution to this Riccati cquation. Usc your result from part (a) to find a
lincar equation satisficd by v. (Answer: v+(-6-g(t))v b))
d) Solve thc differential cquation that you derived in part (c) for the function v in the special case
g(t) = at. Lcave your solution in terms of an integral.
(Answer: v(t) = elbt+) S bel-bt-)dt + cet+)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc740e8b2-d410-40ba-b674-1beb1e9bb454%2F8aa6ccb5-322c-4504-a908-a2b5232001d9%2F9vak9dh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:arc of considerab.c nterest tom
cquation has the form
ns. A Riccati
dy
= A(r)y+ B(x)y + C(x).
dr
The mcthod of solution was discovercd by Eulcr.
a) Show that if y1 (x) is a solution to the Riccati cquation then the substitution y= y+ transforms
1.
the cquation into the first order lincar cquation
dv
+ (B+2AY1)v = -A.
dx
b) Use the method of part (a) to find the general solution for the Riccati cquation
dy
y? - 2ry + 1 + 1
dx
given that y(r)
= z is a solution to this cquation. (Answer: y(1) = 1+ (c-r))
The following application of Riccati cquations is due to Z. Hearon (1952). The propagation of a
single action in a large population (drivers turning on hcadlights for example) often depends partly
on cxtcrnal circumstances (gathering darkncss) and partly on a tendency to imitate others who
have alrcady performed the action. In this case, Hearon argucd that the proportion y(t) who have
performed the action can be described by the cquation
y (t) = (1– y)[g(t) + by].
whcre g(t) mcasurcs the cxtcrnal stimulus and b is the imitation cocfficient.
c) Notc that yı (t) = 1 is a solution to this Riccati cquation. Usc your result from part (a) to find a
lincar equation satisficd by v. (Answer: v+(-6-g(t))v b))
d) Solve thc differential cquation that you derived in part (c) for the function v in the special case
g(t) = at. Lcave your solution in terms of an integral.
(Answer: v(t) = elbt+) S bel-bt-)dt + cet+)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
