I don't understand why s0= c1. Can you please explain it to me?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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I don't understand why s0= c1. Can you please explain it to me?
![**Summary**
The Cauchy-Euler Differential Equation is depicted as:
\[ r^2R'' + rR' - \lambda R = 0 \]
Another key equation with boundary conditions is:
\[ S'' + \lambda S = 0 \]
where
\[ S(0) = 0 \]
\[ S'(\pi) = 0 \]
**Given** \(\lambda\), if \(R(r)\) and \(S(\theta)\) are solutions, then \( U(r, \theta) = R(r)S(\theta) \) is a solution to Laplace's Equation with conditions \( \textcircled{A} \) and \( \textcircled{B} \).
**Note:** \(\lambda\) has non-trivial solutions only for certain values.
**Case 1:** \(\lambda = 0\)
The solutions to the above equations are:
1. \( S_0(\theta) = C_1 \) - Solution to \(\textcircled{X}\)
2. \( R_0(r) = C_2 + C_3 \ln r \) - Solution to the Cauchy-Euler
By setting \( C_3 = 0 \), we want the solution to be bounded as \( r \to 0 \).
Thus, \( U_0(r, \theta) = R_0(r)S_0(\theta) = A_0 \) is a solution to Laplace's Equation with conditions \( \textcircled{A} \) and \( \textcircled{B} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca708747-373f-4dbe-b127-10492ca0e68f%2F2549c272-ab6a-43f0-9295-3cc0f908a166%2F4z9evg_processed.png&w=3840&q=75)
Transcribed Image Text:**Summary**
The Cauchy-Euler Differential Equation is depicted as:
\[ r^2R'' + rR' - \lambda R = 0 \]
Another key equation with boundary conditions is:
\[ S'' + \lambda S = 0 \]
where
\[ S(0) = 0 \]
\[ S'(\pi) = 0 \]
**Given** \(\lambda\), if \(R(r)\) and \(S(\theta)\) are solutions, then \( U(r, \theta) = R(r)S(\theta) \) is a solution to Laplace's Equation with conditions \( \textcircled{A} \) and \( \textcircled{B} \).
**Note:** \(\lambda\) has non-trivial solutions only for certain values.
**Case 1:** \(\lambda = 0\)
The solutions to the above equations are:
1. \( S_0(\theta) = C_1 \) - Solution to \(\textcircled{X}\)
2. \( R_0(r) = C_2 + C_3 \ln r \) - Solution to the Cauchy-Euler
By setting \( C_3 = 0 \), we want the solution to be bounded as \( r \to 0 \).
Thus, \( U_0(r, \theta) = R_0(r)S_0(\theta) = A_0 \) is a solution to Laplace's Equation with conditions \( \textcircled{A} \) and \( \textcircled{B} \).
Expert Solution
Step 1
Given that
(1)
Since we know that
If , then
Laplace transform of double derivative is
(2)
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