y" — 2roy' + r²y = 0 -) Calculate the Wronskian W[1, 2] (t). ) Find the function K(s, t) so that the solution to is given by: y" - 2roy' + ry = g(t), y(0) = y'(0) = 0 So K(s, t)g(s)ds (Hint: see Theorem 3.6.1 from the textbook) :) Find the solution to y" − 2roy' + rỗy = |t|, y(0) = y'(0) = 0
y" — 2roy' + r²y = 0 -) Calculate the Wronskian W[1, 2] (t). ) Find the function K(s, t) so that the solution to is given by: y" - 2roy' + ry = g(t), y(0) = y'(0) = 0 So K(s, t)g(s)ds (Hint: see Theorem 3.6.1 from the textbook) :) Find the solution to y" − 2roy' + rỗy = |t|, y(0) = y'(0) = 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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Question
theorem here is the variation of parameter
![2. The functions ₁(t) = erot, ₂(t) = terot give a fundamental set of solutions to the ODE
y" − 2roy' + r²y = 0
(a) Calculate the Wronskian W[1, 2] (t).
(b) Find the function K(s, t) so that the solution to
is given by:
y" − 2roy' + r²y = g(t), _y(0) = y'(0) = 0
*K(s, t)g(s)ds
0
(Hint: see Theorem 3.6.1 from the textbook)
(c) Find the solution to
y" − 2roy' + r²y = |t|,_y(0) = y'(0) = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24d2d084-e1b8-4965-aed3-95c7dd8fa933%2F3daed2c8-b6cd-4f33-a4bb-58954ffc65a4%2Fsikazym_processed.png&w=3840&q=75)
Transcribed Image Text:2. The functions ₁(t) = erot, ₂(t) = terot give a fundamental set of solutions to the ODE
y" − 2roy' + r²y = 0
(a) Calculate the Wronskian W[1, 2] (t).
(b) Find the function K(s, t) so that the solution to
is given by:
y" − 2roy' + r²y = g(t), _y(0) = y'(0) = 0
*K(s, t)g(s)ds
0
(Hint: see Theorem 3.6.1 from the textbook)
(c) Find the solution to
y" − 2roy' + r²y = |t|,_y(0) = y'(0) = 0
¹0
- ds + y₂(t)
as prescribed by Theorem 3.5.2.
S
y₁(s)g(s)
W[y₁, y2](s)
where to is any conveniently chosen point in I. The general solution is
y = C₁y₁(t) + C2y2(t) +Y(t),
ds,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24d2d084-e1b8-4965-aed3-95c7dd8fa933%2F3daed2c8-b6cd-4f33-a4bb-58954ffc65a4%2Feb2iml_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the nonhomogeneous second-order linear differential equation
y" +p(t)y' +q(t) y = g(t).
If the functions p, q, and g are continuous on an open interval I, and if the functions У1 and Y2
a fundamental set of solutions of the corresponding homogeneous equation
y" + p(t)y' +q(t) y = 0,
then a particular solution of equation (28) is
Y(t) = −y₁(t)
y2(s) g(s)
W[y₁, y2](s)
¹0
- ds + y₂(t)
as prescribed by Theorem 3.5.2.
S
y₁(s)g(s)
W[y₁, y2](s)
where to is any conveniently chosen point in I. The general solution is
y = C₁y₁(t) + C2y2(t) +Y(t),
ds,
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