Consider the differential equation y" + ay' = f(x), where f(x) is an arbitrary given function, a is a real scalar, and the primes indicate deriva- tives with respect to x. We wish to find a general solution y(x) to this equation in [0, ∞) under the initial value boundary conditions y(0) = 0 and y'(0) = 0. i) Show that the Green's function G(x, t) for this differential equation under the given boundary conditions is given by G(x, t) = 0 (1 - exp[-a(xt)]) 0 ≤ x < t t < x. ii) Using this Green's function, write down the solution to the given ODE for arbitrary f(x). iii) Hence find the general solution given f(x) = e-Br Hint: You may want to consider the cases a = ß and a # ß separately.
Consider the differential equation y" + ay' = f(x), where f(x) is an arbitrary given function, a is a real scalar, and the primes indicate deriva- tives with respect to x. We wish to find a general solution y(x) to this equation in [0, ∞) under the initial value boundary conditions y(0) = 0 and y'(0) = 0. i) Show that the Green's function G(x, t) for this differential equation under the given boundary conditions is given by G(x, t) = 0 (1 - exp[-a(xt)]) 0 ≤ x < t t < x. ii) Using this Green's function, write down the solution to the given ODE for arbitrary f(x). iii) Hence find the general solution given f(x) = e-Br Hint: You may want to consider the cases a = ß and a # ß separately.
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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Question
![a)
Consider the differential equation
y" + ay = f(x),
where f(x) is an arbitrary given function, a is a real scalar, and the primes indicate deriva-
tives with respect to x. We wish to find a general solution y(x) to this equation in [0, ∞)
under the initial value boundary conditions y(0) = 0 and y'(0) = 0.
i) Show that the Green's function G(x, t) for this differential equation under the given
boundary conditions is given by
G(x, t) =
(1 - exp[-a(x - t)])
0 ≤ x < t
t < x.
ii) Using this Green's function, write down the solution to the given ODE for arbitrary f(x).
iii) Hence find the general solution given f(x) = e-ßr.
Hint: You may want to consider the cases a = ß and a 3 separately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6ad4c45-0eaf-424e-8409-2b74ccd88e4a%2F26583f00-19b5-437d-aa76-ebfa5645ceac%2Fqtskwt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a)
Consider the differential equation
y" + ay = f(x),
where f(x) is an arbitrary given function, a is a real scalar, and the primes indicate deriva-
tives with respect to x. We wish to find a general solution y(x) to this equation in [0, ∞)
under the initial value boundary conditions y(0) = 0 and y'(0) = 0.
i) Show that the Green's function G(x, t) for this differential equation under the given
boundary conditions is given by
G(x, t) =
(1 - exp[-a(x - t)])
0 ≤ x < t
t < x.
ii) Using this Green's function, write down the solution to the given ODE for arbitrary f(x).
iii) Hence find the general solution given f(x) = e-ßr.
Hint: You may want to consider the cases a = ß and a 3 separately.
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