1. (10 points) Using the definition of Laplace transform, find L{-6t + 11} XXXXXXXXXXpoints) Use the...

Question

1. (10 points) Using the definition of Laplace transform, find L{-6t + 11} XXXXXXXXXXpoints) Use the Laplace transform to solve the following initial value prob-lem. No credit will be given for any other method y/ yt — 6y = —6t + 11, y(0) = —1, '00) = 4
3. (20 points) For each of the following boundary value problems, either show the problem has no solution, find its unique solution, or find all of its infinitely many solutions. (a) yil + y = x + 1. y(0) = 1, y =- (b) yfi + y = x + 1, y(7r) = 0, y(-77) = 0 (c) yti + y = x + 1, y(7r) = 0, y(277) = 3r XXXXXXXXXXpoints) Sketch three periods of the function defined below and find its Fourier series.
f (x)
x 0
Are there any values of x for which the series you found does not converge to the value dictated by the definition of f found above? To what value does the series converge in this case?
(25 points) Sketch three periods of the function defined below and find its Fourier series.
f(x) f(x ± 2) f(x) --{ i_x oIs this function even, odd, or neither? Explain your answer

005_ahdlpk-fk1vr1dk.pdf

Robert · Accepted Answer

1. Using the definition of the Laplace transform, find  6 11 .t L  
 
We have
   
0
0 0
2
6 11 6 11
6 11
6 11
.
st
st st
t t e dt
te dt e dt
s s


 
 
    
  
  

 
L

2. Use the Laplace transform to solve the following initial value problem. 
 
   '' ' 6 6 11; 0 1; ' 0 4.y y y t y y         
 
 We have
   
2
'' ' 6 6 11
6 11
.
y y y t
s s
    
  
L L

 Now we have
       
           
         
      
   
   
2
2
2
2
2
2
'' ' 6 '' ' 6
0 ' 0 0 6
6 1 0 ' 0
6 1 1 4
6 1 4
6 5
6 11
.
y y y y y y
s Y s sy y sY s y Y s
s s Y s s y y
s s Y s s
s s Y s s
s s Y s s
s s
    
       
     
      
     
    
  
L L L L

 Thus we have
 
  
  
  
 
2 2
3 2
2
2
2
2
2
2
1 11 6
5
6
5 11 6
.
3 2
2 7 3
3 2
7 3
3
,
3
Y s s
s s s s
s s s
s s s
s s s
s s s
s s
s s
A B C
s s s
 
     
   
  
 
 
  
 
 
  


  


where A, B, and C are constants. Thus we have
   
   
   
2 2
2 2
2
7 3 3 3
3 3
3 3 .
s s As s B s Cs
A s s B s Cs
A C s A B s B
       
    
     

Equating corresponding coefficients, we obtain the following linear system:
1;
3 7;
3 3.
A C
A B
B
  
  
  

The last equation yields
 1.B   
 
Plugging this value into the second equation, we obtain
 3 1 7,A  
whence
 3 6A   
 
and hence
 2.A 
Finally, plugging this value of A into the first equation, we obtain
 2 1,C   
whence
 1.C   
 
Thus we have
   2
2 1 1
.
3
Y s
s s s
   


Taking the inverse Laplace transform of  Y s , we obtain 
 
   32 .ty t t e   
3. For each of the following boundary value problems, either show the problem has no 
solution, find its unique solution, or find all of its infinitely many solutions.
(a)    2 2'' 1; 0 1; .y y x y y
     
First we find the general solution to the above differential equation. We have
 ,

1. (10 points) Using the definition of Laplace transform, find L{-6t + 11} XXXXXXXXXXpoints) Use the Laplace transform to solve the following initial value prob-lem. No credit will be given for any...

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