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

Part 3

Find $y_{n}(x)\!$ , for $n=4,7,11\!$ such that:

y_{n}''-3y_{n}'+2y_{n}=r_{n}(x)\!

for $x\!$ in $[0.9,3]\!$ with the initial conditions found.

Plot $y_{n}(x)\!$ for $n=4,7,11\!$ for $x\!$ in $[0.9,3]\!$ .

Homogeneous Solution

The homogeneous case is shown below:

y''_{h}-3y'_{h}+2y_{h}=0\!

This equation has the following roots:

$\lambda _{1}=1,\lambda _{2}=2\!$

Which gives yields the homogeneous solution

y_{h}=c_{1}e^{x}+c_{2}e^{2x}\!

General Solution, n=4

Using the taylor series approximation from earlier with $n=4\!$ we have

$r_{4}(x)=log(2)+{\frac {(x-1)}{2ln(10)}}-{\frac {(x-1)^{2}}{8ln(10)}}+{\frac {(x-1)^{3}}{24ln(10)}}-{\frac {(x-1)^{4}}{64ln(10)}}\!$

We know the particular solution, $y_{p4}(x)\!$ , ve will have this form:

$y_{p4}(x)=a_{4}(x-1)^{4}+a_{3}(x-1)^{3}+a_{2}(x-1)^{2}+a_{1}(x-1)+a_{0}\!$

taking the derivatives of this solution

$y'_{p4}(x)={\frac {d}{dx}}y_{p4}(x)=4a_{4}(x-1)^{3}+3a_{3}(x-1)^{2}+2a_{2}(x-1)+a_{1}\!$

and

$y''_{p4}(x)={\frac {d}{dx}}y'_{p4}(x)=12a_{4}(x-1)^{3}+6a_{3}(x-1)^{2}+2a_{2}\!$

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

{\begin{bmatrix}2&0&0&0&0\\-12&2&0&0&0\\12&-9&2&0&0\\0&6&-6&2&0\\0&0&2&-3&2\end{bmatrix}}*{\begin{bmatrix}a_{4}\\a_{3}\\a_{2}\\a_{1}\\a_{0}\end{bmatrix}}={\begin{bmatrix}-{\frac {1}{64ln(10)}}\\{\frac {1}{24ln(10)}}\\-{\frac {1}{8ln(10)}}\\{\frac {1}{2ln(10)}}\\log(2)\end{bmatrix}}\!

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

$a_{4}=-.0034,a_{3}=-.0113,a_{2}=-.0577,a_{1}=-.1624,a_{0}=.1624\!$

So the particular solution $y_{p4}\!$ is

$y_{p4}=0.1624-0.1624*(x-1)-.0577*(x-1)^{2}-.0113*(x-1)^{3}-.0034*(x-1)^{4}\!$

We can now find the general solution for n=4, $y_{4}(x)\!$ .

$y_{4}(x)=y_{h}(x)+y_{p4}(x)\!$

$y_{4}(x)=c_{1}e^{x}+c_{2}e^{2x}+0.1624-0.1624*(x-1)\!$ $-.0577*(x-1)^{2}-.0113*(x-1)^{3}-.0034*(x-1)^{4}\!$

Solving using the initial conditions yields;

         $y_{4}(x)=.0595e^{x}-.0076e^{2x}+0.1624-0.1624*(x-1)\!$ 
           $-.0577*(x-1)^{2}-.0113*(x-1)^{3}-.0034*(x-1)^{4}\!$

General Solution, n=7

Using the taylor series approximation from earlier with $n=7\!$ we have

$r_{7}(x)=log(2)+{\frac {(x-1)}{2ln(10)}}-{\frac {(x-1)^{2}}{8ln(10)}}+{\frac {(x-1)^{3}}{24ln(10)}}-{\frac {(x-1)^{4}}{64ln(10)}}+{\frac {(x-1)^{5}}{160ln(10)}}-{\frac {(x-1)^{6}}{384ln(10)}}+{\frac {(x-1)^{7}}{896ln(10)}}\!$

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

{\begin{bmatrix}2&0&0&0&0&0&0&0\\-21&2&0&0&0&0&0&0\\42&-18&2&0&0&0&0&0\\0&30&-15&2&0&0&0&0\\0&0&20&-12&2&0&0&0\\0&0&0&12&-9&2&0&0\\0&0&0&0&6&-6&2&0\\0&0&0&0&0&2&-3&2\end{bmatrix}}*{\begin{bmatrix}a_{7}\\a_{6}\\a_{5}\\a_{4}\\a_{3}\\a_{2}\\a_{1}\\a_{0}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{896ln(10)}}\\-{\frac {1}{384ln(10)}}\\{\frac {1}{160ln(10)}}\\-{\frac {1}{64ln(10)}}\\{\frac {1}{24ln(10)}}\\-{\frac {1}{8ln(10)}}\\{\frac {1}{2ln(10)}}\\log(2)\end{bmatrix}}\!

Solving:

$a_{7}=.0002,a_{6}=.0020,a_{5}=.0141,a_{4}=.0725,a_{3}=.3034,a_{2}=.9029,a_{1}=1.9072,a_{0}=2.1084\!$

So the particular solution $y_{p7}\!$ is

$y_{p7}=2.1084+1.9072*(x-1)+.9029*(x-1)^{2}+.3034*(x-1)^{3}+.0725*(x-1)^{4}+.0141*(x-1)^{5}+.0020*(x-1)^{6}+.0002*(x-1)^{7}\!$

We can now find the general solution for n=7, $y_{7}(x)\!$ .

$y_{7}(x)=y_{h}(x)+y_{p7}(x)\!$

$y_{7}(x)=c_{1}e^{x}+c_{2}e^{2x}+2.1084+1.9072*(x-1)+.9029*(x-1)^{2}+\!$

$.3034*(x-1)^{3}+.0725*(x-1)^{4}+.0141*(x-1)^{5}+.0020*(x-1)^{6}+.0002*(x-1)^{7}\!$

Solving using our initial conditions yields

     $y_{7}(x)=-.7271e^{x}+.0233e^{2x}+2.1084+1.9072*(x-1)+\!$    
  $.9029*(x-1)^{2}+.3034*(x-1)^{3}+.0725*(x-1)^{4}+.0141*(x-1)^{5}+\!$ 
                $.0020*(x-1)^{6}+.0002*(x-1)^{7}\!$

General Solution, n=11

Using the taylor series approximation from earlier with $n=11\!$ we have

$r_{11}(x)=log(2)+{\frac {(x-1)}{2ln(10)}}-{\frac {(x-1)^{2}}{8ln(10)}}+{\frac {(x-1)^{3}}{24ln(10)}}-{\frac {(x-1)^{4}}{64ln(10)}}+{\frac {(x-1)^{5}}{160ln(10)}}-{\frac {(x-1)^{6}}{384ln(10)}}+\!$

${\frac {(x-1)^{7}}{896ln(10)}}-{\frac {(x-1)^{8}}{2048ln(10)}}+{\frac {(x-1)^{9}}{4608ln(10)}}-{\frac {(x-1)^{10}}{10240ln(10)}}+{\frac {(x-1)^{11}}{22528ln(10)}}\!$

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

$A*{\begin{bmatrix}a_{11}\\a_{10}\\a_{9}\\a_{8}\\a_{7}\\a_{6}\\a_{5}\\a_{4}\\a_{3}\\a_{2}\\a_{1}\\a_{0}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{22528ln(10)}}\\-{\frac {1}{10240ln(10)}}\\{\frac {1}{4608ln(10)}}\\-{\frac {1}{2048ln(10)}}\\{\frac {1}{896ln(10)}}\\-{\frac {1}{384ln(10)}}\\{\frac {1}{160ln(10)}}\\-{\frac {1}{64ln(10)}}\\{\frac {1}{24ln(10)}}\\-{\frac {1}{8ln(10)}}\\{\frac {1}{2ln(10)}}\\log(2)\end{bmatrix}}\!$

Solving the system in matlab:

$a_{11}=0,a_{10}=.0002,a_{9}=.0019,a_{8}=.0181,a_{7}=.15,a_{6}=1.0675,a_{5}=6.4597,\!$

$a_{4}=32.4318,a_{3}=130.0033,a_{2}=390.3968,a_{1}=781.289,a_{0}=781.6873\!$

So the particular solution $y_{p11}\!$ is

$y_{p11}=781.6873+781.289*(x-1)+390.3968*(x-1)^{2}+130.0033*(x-1)^{3}+32.4318*(x-1)^{4}+\!$

$6.4597*(x-1)^{5}+1.0675*(x-1)^{6}+.15*(x-1)^{7}+.0181*(x-1)^{8}+.0019*(x-1)^{9}+.0002*(x-1)^{10}\!$

We can now find the general solution for n=11, $y_{1}1(x)\!$ .

$y_{11}(x)=y_{h}(x)+y_{p11}(x)\!$

$y_{11}(x)=c_{1}e^{x}+c_{2}e^{2x}+781.6873+781.289*(x-1)\!$

$+390.3968*(x-1)^{2}+130.0033*(x-1)^{3}+32.4318*(x-1)^{4}\!$

$6.4597*(x-1)^{5}+1.0675*(x-1)^{6}+.15*(x-1)^{7}+.0181*(x-1)^{8}+.0019*(x-1)^{9}+.0002*(x-1)^{10}\!$

Solving using our initial conditions yields

     $y_{11}(x)=-287.5907e^{x}+.05e^{2x}+781.6873+781.289*(x-1)\!$ 
     
       $+390.3968*(x-1)^{2}+130.0033*(x-1)^{3}+32.4318*(x-1)^{4}\!$ 
      
                $6.4597*(x-1)^{5}+1.0675*(x-1)^{6}+\!$  
   
     $.15*(x-1)^{7}+.0181*(x-1)^{8}+.0019*(x-1)^{9}+.0002*(x-1)^{10}\!$

Plot

$y_{4}\!$ shown in red

$y_{7}\!$ shown in blue

$y_{11}\!$ shown in green

Part 4

Solved by Luca Imponenti

Use the matlab command ode45 to integrate numerically $y''-3y'+2y=r(x)\!$ with $r(x)=log(1+x)\!$

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

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

Plot

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)