University of Florida/Egm4313/s12.team12.R2

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Report 2

Problem 2.1[edit]

Given[edit]

For part one find the non-homogeneous L2-ODE-CC in standard form and the solution in terms of the initial conditions and the general excitation

Consider no excitation and plot the solution

For part two you need to generate 3 non-standard (and non-homogeneous) L2-ODE-CC that admit the two values as the two roots of the corresponding characteristic equation.


Solution[edit]

Find characteristic equation

The homogeneous solution takes the form

Take the derivative and subitute the intial conditions into the equations

The Solution to the non homogeneous L2-ODE


Part two:

Characteristic Equation

Multiply equation by a variable, i chose m = 2,4,6 respectfully


Author[edit]

Patrick Greivell, 15:45 8 February 2012 (UTC)

Problem 2.2[edit]

Given[edit]

Given is an Initial Value Problem (IVP) to solve a second-order Ordinary Differential Equation (ODE) by finding a particular solution to the equation and its subsequent plot.

Initial Values:

4313team12r22-13.jpg

4313team12r22-14.jpg

Differential Equation:

4313team12r22-16.jpg

Assume that no excitation exists, i.e.

4313team12r22-15.jpg

Therefore the ODE to be solved becomes:

4313team12r22-17.jpg

Solution[edit]

With the given conditions, a homogenous linear second order ODE with constant coefficients is given to solve. In order to solve this ODE, a basis of solutions must be formed that meet the given conditions. Our goal is to find a solution to that will fit to the a general solution in the form of:

4313team12r11-29.jpg

To do this, a quadratic equation in the form of the following is used to determine the method of how the basis of solutions will be formed.

4313team12r11-18.jpg

In order to do find the general solution, and ultimately the particular solution for this IVP, we will need to solve for λ using the characteristic equation above. Given that this is a quadratic equation, we can solve for λ by finding the roots of the equation. We know the variables a and b from our given ODE.

Let:

a = -10 b = 25

Plugging this into the characteristic equation, we yield the following:

4313team12r11-19.jpg

Now, instead of solving directly for λ, we can determine exactly how many and what kind of roots the equation will have based on the following cases given the value of the discriminant:

Case 1 Case 2 Case 3
Discriminant > 0 Discriminant = 0 Discriminant < 0
Two Real Roots One Real Root Two Complex Roots

The discriminant is given by the following equation:

4313team12r11-30.jpg

If we evaluate the discriminant using the values of a and b from above, we can evaluate the discriminant to be:

4313team12r11-20.jpg

Therefore, we are in Case 2, where the value of our roots will be one real double root. Therefore, we can determine the value of λ by the following equation and by substituting the values into the equation subsequently.

4313team12r11-31.jpg

4313team12r11-21.jpg

4313team12r11-22.jpg

From this, we can use the value of λ to form one equation of the basis as follows:

4313team12r11-23.jpg

In order to form a basis, a second solution is needed. This is done by performing a reduction of order, yielding a first order differential equation. This process will produce the following general solution*:

4313team12r11-32.jpg

Substituting the value for λ that was found:

4313team12r11-24.jpg

In order to solve for the variables c1 and c2, we will use the initial conditions that were given in the problem statement. Therefore, we have two unknowns. We can solve for both of these variables by writing two equations, since one of the initial values was given as a value for the first derivative of the solution at a given point. The derivative of the general solution obtained (shown directly above) is obtained:

4313team12r11-25.jpg

We can determine the value of c2 in terms of c1 by simply substituting in the known initial values as follows:

4313team12r11-26.jpg

The second equation to solve for the unknown variable c1 can be determined by simply using the general solution equation and using the other initial value in the problem statement.

4313team12r11-27.jpg

Therefore, we now know both of the unknown variables. Therefore the particular solution of the given ODE is:

4313team12r11-28.jpg

It's subsequent plot is therefore:

File:4313team12r11plot.jpg

Author[edit]

--Egm4313.s12.team12.stewart 04:35, 6 February 2012 (UTC)

  • Proof for form of general solution found in: Kreyszig, Erwin, Herbert Kreyszig, and E. J. Norminton. Advanced Engineering Mathematics. Hoboken, NJ: Wiley, 2011. Print.
  • Plot for particular solution plotted using Wolfram Alpha

Problem 2.3[edit]

Statement[edit]

K 2011 p.59 pbs. 3-4
Find a general solution. Check your answer by substitution.

Solution[edit]

Problem 3:
Consider an ODE of the form:



First we notice that the ODE is homogenous, linear, and has constant coefficients.
Therefore, we find the auxiliary equation, which is of the form:



The value of the discriminant is:


We plug in a = 6 and b = 8.96 into the auxillary equation to find the roots.



The general solution will be of the form:



Therefore the solution will be




Problem 4:
Consider an ODE of the form:



First we notice that the ODE is homogenous, linear, and has constant coefficients.
Therefore, we find the auxiliary equation, which is of the form:



Looking at the coefficients we obtain:



Next we obtain the roots of this equation with the quadratic formula:



This can be simplified to:





We notice that by looking inside the radical, we will be obtaining complex roots. Continuing with simplification we get:



The general solution to the case of complex roots is:



Where complex roots are of the form:



Knowing and , we get the final general solution:



Author[edit]

Egm4313.s12.team12.sutcliffe 18:42, 8 February 2012 (UTC)

Problem 2.4[edit]

Statement[edit]

K 2011 p.59 pbs. 5-6
Find a general solution. Check your answer by substitution.

Solution[edit]

Problem 5:
Consider an ODE of the form:



This ODE is linear, and has constant coefficients. Therefore, its general solution is of the form:



Therefore:



And:



The ODE can be rearranged so that:



And since:



Then:







Since:



The general solution then becomes:





Therefore:



Problem 6:
Consider an ODE of the form:



Divide through by 10 to get the equation into standard form where the leading coefficient is equal to 1.



First we notice that the ODE is homogenous, linear, and has constant coefficients.
Therefore, we find the auxiliary equation, which is of the form:



The characteristic equation then becomes



The value of the discriminant is:



The 0 indicates that there is a real double root.

The general solution will be of the form:



Where -a/2 = -(-3.2)/2 = 1.6

Therefore the solution will be



Author[edit]

Egm4313.s12.team12.sutcliffe 18:43, 8 February 2012 (UTC)

Problem 2.5[edit]

Given[edit]


K 2011 p.59 pbs. 16-17
Find an ODE



for the given basis:




Solution[edit]

Problem 16.
Distinct real roots are of the form:



Therefore,

The characteristic equation is:



Therefore we have:


Through simplification and distribution we get:


This is the auxiliary equation to the original linear, homogeneous ODE.
Therefore our final solution is:



Problem 17.
Double real roots are of the form:



Therefore,

The characteristic equation is:



This is the auxiliary equation to the original linear, homogeneous ODE.
Therefore our final solution is:



Author[edit]

Egm4313.s12.team12.stadick 24:00, 7 February 2012 (UTC)

Problem 2.6[edit]

Given[edit]

R1.3 image.png


Find the parameters k, c, m of the the spring-dotplot-mass system with a double root


Solution[edit]

The equation of motion for this system is

or

The homogeneous equation is

Characteristic Equation of the double root

Then you find the homogeneous L2-ODE-CC for the characteristic eq.

Now compare the coeffients with the motion equation



Author[edit]

Patrick Greivell, 16:16 8 February 2012(UTC)

Problem 2.7[edit]

Given[edit]

Develop the Maclaurin Series (Taylor Series @ t = 0) for

a)

b)

c)

Solution[edit]

The General form of a Taylor Series is

with the expansion at point a being


The Maclaurin Series (Taylor Series @ t=0) would be

a)

which would equal

or

for all x



b)

which would equal

or

for all x



c)

which would equal

or

for all x


Author[edit]

Cpettigrew 21:15, 5 February 2012 (UTC)

Problem 2.8 (a)[edit]

Given[edit]

#8 P. 59 Given find a general solution and check your answer by substitution.

Solution[edit]





Since it is a case where the solutions are complex conjugate roots.

A general solution for an ODE of this case can be given by the equation

1st Step

The 1st step to finding the solutions for this ODE is to substitute the variable in for y and its derivatives.

and

So becomes

2nd Step

Now that the ODE has been made into a quadratic equation of form we can use the quadratic formula

Inputting the coefficients from the equation and solving for you get

or and

where and

so or

3rd Step

Now if you plug all of this in to the general solution you will get

Checking Solution with Substitution

Using the general solution and taking the first and second derivatives of this general solution we can plug that in to the original equation and see if we in fact get zero. If so, then the general solution that was found is a solution for the ODE.



The first derivative is

The second derivative is



Plugging all of this into the original equation you get







Simplifying all of these terms you end up getting



Author[edit]

--Egm4313.s12.team12.anders. 4:11, 7 February 2012 (UTC)

Problem 2.8 (b)[edit]

Given[edit]

#15 P.59 Given find a general solution and check your answer by substitution.

Solution[edit]





Since it is a case where the solutions are complex conjugate roots.

A general solution for an ODE of this case can be given by the equation

1st Step

The 1st step to finding the solutions for this ODE is to substitute the variable in for y and its derivatives.

and

So becomes

2nd Step

Now that the ODE has been made into a quadratic equation of form we can use the quadratic formula

Inputting the coefficients from the equation and solving for you get

or and

where and

so or

3rd Step

Now if you plug all of this in to the general solution you will get

Checking Solution with Substitution

Using the general solution and taking the first and second derivatives of this general solution we can plug that in to the original equation and see if we in fact get zero. If so, then the general solution that was found is a solution for the ODE.



The first derivative is

The second derivative is



Plugging all of this into the original equation you get







Simplifying all of these terms you end up getting



Author[edit]

--Egm4313.s12.team12.anders. 4:11, 7 February 2012 (UTC)

Problem 2.9[edit]

Given[edit]

The ODE is:



The given initial conditions are:







The corresponding characteristic equation is:



Solution[edit]

Solving for :











Therefore, the general solution will be:



Where:



And:



Thus:



Therefore:











Final Solution is as follows:





Team12report2problem9figure1.gif


Author[edit]

Wagner Schulz 13:00, 8 February 2012 (UTC)

Contribution Table[edit]

Problem Number Assigned To Solution By Proofread By
2.1 Example Example Example
2.2 Daniel Stewart Daniel Stewart James Stadick
2.3 Garrett Sutcliffe Garrett Sutcliffe Example
2.4 Garrett Sutcliffe Garrett Sutcliffe Example
2.5 James Stadick James Stadick Example
2.6 Example Example Example
2.7 Craig Pettigrew Craig Pettigrew Example
2.8 Lauren Anders Lauren Anders Example
2.9 Wagner Schulz Wagner Schulz Example