So part (b) applied at
gives
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Chapter 5.4, Problem 19E
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To solve (1) proceed as follows;
We need to solve
Since,
Chapter 6.1, Problem 5E
in
with
vanishing on
the radial Laplacian equation;
Implies;
(2)
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Step 3 of 3
Let
in equation (2), to obtain;
Mu
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Chapter 6.1, Problem 5E
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Chapter 5.6, Problem 8E
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Or;
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Chapter 6.1, Problem 5E(4)
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Integrate (4) with respect to r, to obtain;
(5)
Where,
is an arbitrary constant.
Observe that the term
introduces a singularity at
, therefore a
Therefore, by the method of shifting the data, put
Then,
as
(1)
and
Also,
And,
Therefore, the problem now becomes;
(2)
Let the solution of (1) be of the form
(3)
With boundary conditions
Then, substit
Step 3 of 7
First Case: When
, the general solution of
Apply the boundary conditions to obtain a trivial solution. This implies that this case is not
feasible.
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Step 4 of 7
Second Case: When
Th
Step 7 of 7
Substitute values of
in (2) to obtain;
Chapter 5.6, Problem 2E
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(5)
Where
and
Apply the initial condition from (1) in (5) to obtain;
Which represents the Fourier se
Adding them and simplifying, we get
.
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Apply the boundary condition
to the above expression to obtain;
In order to have a non-trivial solution, the constant
must be non-zero and clearly
this implies;
Implies;
Where
Therefore;
Where
is an a
Step 2 of 2
At
, we have
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Chapter 6.3, Problem 3E
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Matching the terms, we have
and all other coefficients are zero. So
.
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Where
is an arbitrary constant and
Substitute values of
in (3) to obtain;
(6)
Where
and
Apply the initial condition from (2) in (6) to obtain;
Which represents the Fourier series and therefore, the Fo
(7)
Hence, from (1) and (7);
Comment
Step 7 of 7
(b)
The equilibrium state is;
As the second term in the expression of
tends to zero as
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, the general solution of
Apply the boundary conditions to obtain a trivial solution. This implies that this case is not
feasible.
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Chapter 5.6, Problem 2E
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Step 4 of 7
Implies;
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Chapter 5.6, Problem 2E
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Where
Therefore;
Where
is an arbitrary constant and
Comment
Step 6 of 7
Similarly, proceeding to solve for T(t), a linear first order ordinary
Equation (3) is same as the Bessels equation of order zero.
Therefore, general solution of the equation (3) is,
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Chapter 6.1, Problem 3E
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Step 5 of 5
Replace the variabl
where
,
Chapter 5.6, hence
Problem
we 5E
get
and
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for all other n.
Comment
Step 2 of 2
Now for the inhomogeneous equations, we get
and
hence
and
,
for all other n.
Therefore, the solution is g
Now let
.
chain 3E
rule to determine the derivatives of
Chapter 6.1, Use
Problem
with respect to
Comment
Step 4 of 5
Substitute the values for
(3)
in the equation (2).
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ON
Step 2 of 3
(b)
Chapter 6.1, Problem 9E
We need to find the hottest and coldest temperatures.
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From part (a), the temperature decreases as r increases. Therefore, its hottest t
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Chapter 5.5, Problem 2E
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Implies;
Comment
Step 4 of 5
Integrate both sides of equation (4) with respect to r, to obtain;
(5)
Use the condition
on
in equation (5), to obtain;
Implies;
(6)
Now substitute the value of
More speci
Therefore,
where
with initial conditions
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Chapter 6.1, Problem 3E
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Kamalkisho
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Chapter 5.6, Problem 5E
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Chapter 6.1, Problem 1E
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Step 2 of 2
Multiplying this equation by
and equating the result with the obtained equation
differentiating with respect to x, we get
Now, we c
As
, so it is enough to find the solutions of the following ordinary differential equation.
. (2)
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Chapter 6.1, Problem 3E
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Step 2 of 5
Consider the following Bessels equation of order
Comments (1)
Step 2 of 2
This simplifies to
or
.
We know that the solution of this equation has the form
.
So, the required solution is
.
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