Exam2_s.pdf - MATH 241 - Partial Differential Equations...

MATH 241 - Partial Differential Equations ( Exam # 2 )
Summer Semester I, 2021 M. Carchidi
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Solving The ODE: ay ′′  x   by ′  x   cy  x   0 ( a , b , c constants with a ≠ 0)
D  b 2 − 4 ac y
1
 x  y
2
 x    
Negative e  x cos   x  e  x sin   x  − b /  2 a  − D /  2 a  −
Zero e
 x
xe
 x
− b /  2 a  − −
Positive e  x cosh   x  e  x sinh   x  − b /  2 a  − D /  2 a 
e      x e   −   x − b /  2 a  − D /  2 a 
y  x   c
1
y
1
 x   c
2
y
2
 x 
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Given the ODE

n
′′  x   
n

n
 x   0
for 0  x  L , we have, for the four basic boundary conditions (BCs), the following table of
results.
Case BCs n 
n

n
 x  
n
 
n
1 
n
 0   
n
 L   0 1,2,3,... n  / L sin  x 
n
 L /2
2 
n
′  0   
n
 L   0 1,2,3,...  n − 1/2   / L cos  x 
n
 L /2
3 
n
 0   
n
′  L   0 1,2,3,...  n − 1/2   / L sin  x 
n
 L /2
4 a 
n
′  0   
n
′  L   0 1,2,3,... n  / L cos  x 
n
 L /2
4 b Also n  0 
0
 0 
0
 x   1 L
where
f  g 

0
L
f  x  g  x  dx and 
p
 
q
 0
when p ≠ q .
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Given the PDE
a
∂ 2 u  x , t 
∂ x 2
 b
∂ 2 u  x , t 
∂ x ∂ t
 c
∂ 2 u  x , t 
∂ t 2
 0
with a , b and c constants, the following table summarizes a general solution for these.
Solving The PDE: au
xx
 x , t   bu
xt
 x , t   cu
tt
 x , t   0 ( a , b , c constants)
Case # Conditions Quadratic Equation Roots u  x , t 
1 c ≠ 0  b
2
− 4 ac ≠ 0  a  br  cr
2
 0 r

, r
−
f  x  r

t   g  x  r
−
t 
2 c ≠ 0  b 2 − 4 ac  0  a  br  cr 2  0 r
0
, r
0
f  x  r
0
t   tg  x  r
0
t 
3 a ≠ 0  b 2 − 4 ac ≠ 0  as 2  bs  c  0 s

, s
−
f  s

x  t   g  s
−
x  t 
4 a ≠ 0  b 2 − 4 ac  0  as 2  bs  c  0 s
0
, s
0
f  s
0
x  t   xg  s
0
x  t 
5 a  c  0  b ≠ 0  − − f  x   g  t 
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2

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Problem # 1 ( 25 Points ) - Solving a PDE With Two ICs and One BC
Determine the unique solution to the following PDE
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Problem # 2 ( 25 Points ) - A General Solution to a Wave - Like PDE and BCs
Construct a general solution to the wave equation
∂ 2 u  x , t 
∂ x 2

∂ 2 u  x , t 
∂ t 2
 2
for 0  x  1 and 0  t , given the boundary conditions
u  0, t   0 and
∂ u  x , t 
∂ x
x  1
 1
for all 0  t . Hint : You may use the above table for the eigenvalue/eigenfunction problem
involving the ODE 
n
′′  x   
n

n
 x   0 in order to save some time.