University of Florida/Egm4313/IEA-f13-team10/R4

Report 4

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Problem 1: Basic Rule to Solve Non-Homogeneous ODE

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Problem Statement

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Solve the ODE:

 

With the initial conditions:


 

 

Plot the homogeneous solution

Plot the particular solution

Plot the overall solution

Solution

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Homogeneous solution:  

So that  

Solving for the initial conditions:

 

 

 
 

So the homogeneous solution is:

 

Choose the particular solution to be:  

So that:

 

 

Substitute in the original equation  

 
 

Sorting by the x term gives:

 
 

Giving us the system of equations:


 
 
 
 
 
 
 
 

The Matlab code we made to solve this was:

A = -2/9;

B = -(42*A)/9;

C = -(42*A + 36*B)/9;

D = -(30*B + 30*C)/9;

E = -(20*C + 24*D)/9;

F= -(12*D + 18*E)/9;

G = (8-(6*E + 12*F))/9;

H = -(2*F + 6*G)/9;

To get (rounded to the nearest tenth):

A= -0.2

B= 1.0

C= -3.1

D= 6.9

E= -11.5

F= 13.8

G= -9.9

H= 3.5

So now the particular solution is:

 

The overall solution can be found by:

 

 

To solve for C_1 and C_2 (which are different from the homogeneous solution constants) we find:

 

   

 
 

The overall solution is:

 

Plot the homogeneous solution:
 

Plot the particular solution:
 

Plot the overall solution:
 

Honor Pledge

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On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 2: Sum Rule to Find Particular Solution

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Problem Statement

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Part 1

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Use the Basic Rule 1 and the Sum Rule 3 to show that the appropriate particular solution for:
 
 

Part 2

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Derive the Basic rule and the Sum rule, instead of just using them, based on the linearity
of the differential operator to obtain the expression (trial solution) for the particular solution  

Solution

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With n =7 we get

 

 

 

From report problem 1 it was already found that:

 
 
 
 
 
 
 
 

 

Honor Pledge

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On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 3: Method of Undetermined Coefficients

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Problem Statement

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Problem Set 2.7, problem 5

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Find a real, general solution. State which rule you are using. Show each step of your work.

 
 

plot on separate graphs:

(1) the homogeneous solution  ,

(2) the particular solution  ,

and (3) the overall solution  .

Solution

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We start by finding the general solution of the homogeneous ODE

 

The characteristic equation of the homogeneous ODE is

 
 
 

The roots are double real roots.
The general solution of the homogeneous ODE is  

Now we solve for the particular solution of the nonhomogeneous ODE

 

We use the method of Undetermined Coefficients

 

 

 
 

We now substitute the values of   into  

 
 
 

 
 
 

Now we equate the coefficients of like terms on both sides
 
 

Now we solve these equations for the coefficients
 
 
 

 
 

These values are substituted into   to get the particular solution of the ODE
 

The general solution of the ODE is
 
 

In order to determine the values of   we use the initial conditions  
 
 

 
 
 
 
 

The general solution of the ODE is
 

Plot the homogeneous solution:
 
Plot the particular solution:
 
Plot the overall solution:
 

Honor Pledge

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On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 4: Method of Undetermined Coefficients

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Problem Statement

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Problem Set 2.7, problem 5

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Find a real, general solution. State which rule you are using. Show each step of your work.

 
 

plot on separate graphs:

(1) the homogeneous solution  ,

(2) the particular solution  ,

and (3) the overall solution  .

Solution

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We start by finding the general solution of the homogeneous ODE

 

The characteristic equation of the homogeneous ODE is

 
 
 

The roots are double real roots.
The general solution of the homogeneous ODE is  

Now we solve for the particular solution of the non-homogeneous ODE

 

By using the definition of hyperbolic trigonometric functions we can convert  .
Our non-homogeneous ODE can now be written as:

 

 

Since the replacement of   is the sum of two functions we can use the sum rule for the method of undetermined coefficients.

 

 

 

 

We now substitute the values of   into  

 
 
 

 

 

Now we equate the coefficients of like terms on both sides and solve for the coefficients
 
 
 

These values are substituted into   to get the particular solution of the ODE

 

The general solution of the ODE is:

 
 

In order to determine the values of   we use the initial conditions  
 
 

 
 
 
 

The general solution of the ODE is
 

Plot the homogeneous solution:
 
Plot the particular solution:
 
Plot the general solution:
 

Honor Pledge

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On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 5: Display of Equality by Series Expansion

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Problem Statement

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Expand the series on both sides of (1)-(2) p.7-12b to verify these equalities
 

 

 

 

Solution

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Evaluating the right-hand side of (1):

 

 

Now evaluating the left-hand side of (1):

 

 

So both sides are equal.

Now evaluating the right-hand side of (2):

 

 

The left-hand side of (2) expands into the following:

 

 

So both sides are equal.

Honor Pledge

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On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 6: Taylor Series to Solve ODE

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Problem Statement

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  and  

Solution

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The taylor series for the excitation is   from n=0 to infinity
For n=3, this equals  
For n=5, this equals  
For n=9, this equals  


For n=3,
 
 
 


Plugging these into the original equation using the taylor series approximation as the excitation,
 
 
Rearranging the coefficients,
 
 


Equating x^6 coefficients, A=-45/4
Equating x^5 coefficients, B=405
Equating x^4 coefficients, C=-118461.833
Equating x^3 coefficients, D=2770183.992
Equating x^2 coefficients, E=-37069430.492
Equating x^1 coefficients, F=-594423101.472
Equating x^0 coefficients, G=4233788358.69
 

The graph shown is the taylor series for cos(2x) for the 0th through 3rd order.

 


Honor Pledge

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On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 7: Taylor Series Expansion of the log Function

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Problem Statement

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Use the point
 
 

Solution

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The taylor series expansion for   around   up to 16 terms is
 
 
 
Plots of taylor series expansion: Up to order 4
 

Up to order 7
 

Up to order 11
 

Up to order 16
 

The visually estimated domain of convergence is from .8 to .2.
Now use the transformation of variable
 
 

If   has a domain of convergence from   then   converges from  

Honor Pledge

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On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.