University of Florida/Egm4313/s12.team11.imponenti/R4.4

Part 3

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Solved by Luca Imponenti

Find   , for   such that:

 

for   in   with the initial conditions found.

Plot   for   for   in  .

Homogeneous Solution
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The homogeneous case is shown below:

 

This equation has the following roots:

 

Which gives yields the homogeneous solution

 
General Solution, n=4
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Using the taylor series approximation from earlier with   we have

 

We know the particular solution,  , ve will have this form:

 

taking the derivatives of this solution

 

and

 

Plugging the above equations into the original ODE yields the following matrix equation:

 

The unknown vector   can be easily solved by forward substitution,the following values were calculated in matlab:

 

So the particular solution   is

 

We can now find the general solution for n=4,  .

 

   

Solving using the initial conditions yields;

         
           
General Solution, n=7
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Using the taylor series approximation from earlier with   we have

 

In a similar fashion we construct a matrix equation for n=7:

 

Solving:

 

So the particular solution   is

 

We can now find the general solution for n=7,  .

 

 

 

Solving using our initial conditions yields

        
  
                
General Solution, n=11
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Using the taylor series approximation from earlier with   we have

 

 

Finally, we write out the matrix equation for n=11:

 

 

Solving the system in matlab:

 

 

So the particular solution   is

 

 

We can now find the general solution for n=11,  .

 

 

 

 

Solving using our initial conditions yields

     
     
       
      
                 
   
     
Plot
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  shown in red

  shown in blue

  shown in green

 

Part 4

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Solved by Luca Imponenti

Use the matlab command ode45 to integrate numerically   with  

and the initial conditions from Part 3 to obtain the numerical solution for y(x).

Plot y(x) in the same figure as above.

Matlab Solution
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The numerical solution calculated using the matlab ode45 command is shown below:

 ans =
   0.2788
   0.2854
   0.2923
   0.2997
   0.3074
   0.3229
   0.3401
   0.3592
   0.3804
   0.4040
   0.4302
   0.4595
   0.4921
   0.5285
   0.5691
   0.6145
   0.6651
   0.7218
   0.7850
   0.8557
   0.9346
   1.0228
   1.1213
   1.2313
   1.3542
   1.4914
   1.6445
   1.8155
   2.0063
   2.2193
   2.4569
   2.7219
   3.0175
   3.3471
   3.7146
   4.1243
   4.5809
   5.0898
   5.6568
   6.2885
   6.9921
   7.3442
   7.7142
   8.1032
   8.5119
Plot
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Plotting the aboved vector of y-values,along with the results from earlier yields the following graph:

 

where the answer calculated in matlab is shown in yellow.

Egm4313.s12.team11.imponenti (talk) 08:04, 14 March 2012 (UTC)