x and time t of a particular heat equation is: U(x,t) = [Cn(e-6tn^2)sin(nx) where the sum goes from 1 to ∞. A) Write out the heat equation that has this solution including the boundary conditions. B) If U(x,0) = 3sin(2x) -5sin(4x) find the two non-zero coefficients Cn C) Find U(x,t).
x and time t of a particular heat equation is: U(x,t) = [Cn(e-6tn^2)sin(nx) where the sum goes from 1 to ∞. A) Write out the heat equation that has this solution including the boundary conditions. B) If U(x,0) = 3sin(2x) -5sin(4x) find the two non-zero coefficients Cn C) Find U(x,t).
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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![A general solution for the temperature, U, as a function of position x and time t of a particular heat equation is:
\[ U(x,t) = \sum_{n=1}^{\infty} C_n e^{-6tn^2} \sin(nx) \]
**A)** Write out the heat equation that has this solution, including the boundary conditions.
**B)** If \( U(x,0) = 3\sin(2x) - 5\sin(4x) \), find the two non-zero coefficients \( C_n \).
**C)** Find \( U(x,t) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee0d03cc-7cc4-4eb9-a594-2d336c0fb45c%2F23321a81-45e2-4184-a64d-e0ca56d72f45%2F5qc5x3j_processed.png&w=3840&q=75)
Transcribed Image Text:A general solution for the temperature, U, as a function of position x and time t of a particular heat equation is:
\[ U(x,t) = \sum_{n=1}^{\infty} C_n e^{-6tn^2} \sin(nx) \]
**A)** Write out the heat equation that has this solution, including the boundary conditions.
**B)** If \( U(x,0) = 3\sin(2x) - 5\sin(4x) \), find the two non-zero coefficients \( C_n \).
**C)** Find \( U(x,t) \).
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