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

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Part 3[edit]

Solved by Luca Imponenti

Find , for such that:

for in with the initial conditions found.

Plot for for in .

Homogeneous Solution[edit]

The homogeneous case is shown below:

This equation has the following roots:

Which gives yields the homogeneous solution

General Solution, n=4[edit]

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[edit]

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[edit]

Using the taylor series approximation from earlier with we have

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

R4.4.JPG

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[edit]

shown in red

shown in blue

shown in green

Plotr4.JPG

Part 4[edit]

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[edit]

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[edit]

Plotting the aboved vector of y-values,along with the results from earlier yields the following graph:

Plotr4ode45.JPG

where the answer calculated in matlab is shown in yellow.

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