4. Suppose that in the plate described in Example 2, Sec. 43, there is a heat source de- pending on the variable y and that the entire boundary is kept at temperature zero. According to Sec. 23, the steady temperatures u(x, y) in the plate must now satisfy Poisson's equation Uxx (x, y) + Uyy(x, y) +q(y) = 0 (a) By assuming a (bounded) solution of the form u(x, y) = u(x, y) n=1 of this temperature problem and using the method of variation of parameters (Sec. 42), show formally that = 8 B₁(x) = B₂ (x) sin ny (n = 1, 2,...), where qn are the coefficients in the Fourier sine series for q(y) on the interval 0 0,0
4. Suppose that in the plate described in Example 2, Sec. 43, there is a heat source de- pending on the variable y and that the entire boundary is kept at temperature zero. According to Sec. 23, the steady temperatures u(x, y) in the plate must now satisfy Poisson's equation Uxx (x, y) + Uyy(x, y) +q(y) = 0 (a) By assuming a (bounded) solution of the form u(x, y) = u(x, y) n=1 of this temperature problem and using the method of variation of parameters (Sec. 42), show formally that = 8 B₁(x) = B₂ (x) sin ny (n = 1, 2,...), where qn are the coefficients in the Fourier sine series for q(y) on the interval 0 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
4
![4. Suppose that in the plate described in Example 2, Sec. 43, there is a heat source de-
pending on the variable y and that the entire boundary is kept at temperature zero.
According to Sec. 23, the steady temperatures u(x, y) in the plate must now satisfy
Poisson's equation
Uxx (x, y) + Uyy(x, y) +q(y) = 0
(a) By assuming a (bounded) solution of the form
u(x, y) =
u(x, y)
n=1
of this temperature problem and using the method of variation of parameters
(Sec. 42), show formally that
=
8
B₁(x) =
B₂ (x) sin ny
(n = 1, 2, ...),
where qn are the coefficients in the Fourier sine series for q(y) on the interval
0 <y<л.
4Q
408
qn
12 (1- e-mx)
(b) Show that when q(y) is the constant function q(y) = Q, the solution in part (a)
becomes
(x>0,0<у<л).
1- exp[-(2n-1)x]
(2n-1)³
sin (2n-1)y.
n=1
Suggestion: In part (a), recall that the general solution of a linear second-order
equation y" + p(x)y g(x) is of the form y=ye+ Yp, where Ур is any particular
solution and ye is the general solution of the complementary equation
=
y" + p(x) y = 0.¹](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F46d4808c-2450-4700-ab8d-04f9f85bf5cf%2Fkj7pr5j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Suppose that in the plate described in Example 2, Sec. 43, there is a heat source de-
pending on the variable y and that the entire boundary is kept at temperature zero.
According to Sec. 23, the steady temperatures u(x, y) in the plate must now satisfy
Poisson's equation
Uxx (x, y) + Uyy(x, y) +q(y) = 0
(a) By assuming a (bounded) solution of the form
u(x, y) =
u(x, y)
n=1
of this temperature problem and using the method of variation of parameters
(Sec. 42), show formally that
=
8
B₁(x) =
B₂ (x) sin ny
(n = 1, 2, ...),
where qn are the coefficients in the Fourier sine series for q(y) on the interval
0 <y<л.
4Q
408
qn
12 (1- e-mx)
(b) Show that when q(y) is the constant function q(y) = Q, the solution in part (a)
becomes
(x>0,0<у<л).
1- exp[-(2n-1)x]
(2n-1)³
sin (2n-1)y.
n=1
Suggestion: In part (a), recall that the general solution of a linear second-order
equation y" + p(x)y g(x) is of the form y=ye+ Yp, where Ур is any particular
solution and ye is the general solution of the complementary equation
=
y" + p(x) y = 0.¹
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