hmk#2_s.pdf - ENM510 - Foundations of Engineering...

ENM510 - Foundations of Engineering Mathematics I ( Homework # 2 )
Fall Semester, 2020 M. Carchidi
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Problem # 1 ( 12 points )
We have seen in class that a general solution to the partial differential equation (PDE)
T
0
∂ 2 w  x , t 
∂ x 2
 
0
∂ 2 w  x , t 
∂ t 2
for 0  t , 0  x  L and the boundary conditions (BCs)
w  0, t   0 and w  L , t   0
for 0 ≤ t , is
w  x , t  
∑
n  1

 A
n
cos  
n
t   B
n
sin  
n
t  
n
 x 
(for arbitrary constants A
n
and B
n
) with

n
 x   sin  
n
x  , 
n

n 
L
and 
n
 
n
T
0

0
.
Using a similar set of steps that lead to this solution, determine a general solution to the
PDE
T
0
∂ 2 w  x , t 
∂ x 2
 
0
∂ 2 w  x , t 
∂ t 2
for 0  t , 0  x  L and the boundary conditions (BCs)
∂ w  x , t 
∂ x
x  0
 0 and w  L , t   0
for 0 ≤ t .
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Problem # 2 ( 12 points )
Using a similar set of steps as in Problem #1, determine a general solution to the PDE
T
0
∂ 2 w  x , t 
∂ x 2
 
0
∂ 2 w  x , t 
∂ t 2
for 0  t , 0  x  L and the boundary conditions (BCs)
w  0, t   0 and
∂ w  x , t 
∂ x
x  L
 0
for 0 ≤ t .
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Problem # 3 ( 20 points )
Using a similar set of steps as in Problem #1, determine a general solution to the PDE
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Problem # 4 ( 12 points )
Using a similar set of steps as in Problem #1, show that a general solution to the partial
differential equation (PDE)
Problem # 5 ( 12 points )
Using a similar set of steps as in Problem #4, determine a general solution to the PDE
Problem # 6 ( 12 points )
Using a similar set of steps as in Problem #4, determine a general solution to the PDE
∂ 2 w  x , t 
∂ x 2

1
k
∂ w  x , t 
∂ t
(for constant k )
for 0  t , 0  x  L and the boundary conditions (BCs)
w  0, t   0 and
∂ w  x , t 
∂ x
x  L
 0 for 0 ≤ t .
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