Finding a particular solution satisfying the initial
Display the general solution below
Define the general solution. Use c1, c2 and c3 ( do
not use subscripts).
Define and compute the third solution.
Define and compute the second solution.
Define and compute the first solution. Use the symbolic
State the condition on v1 for the method to work.
Define v1 using v2 and use the symbolic equal to display
State the condition on v2 for the method to work.
Define v2 using v3 and use the symbolic equal (the
arrow) to display the result.
Define the components of v3. Use subscripts.
Compute the rref of
So, the eigenvalue is defective with defect 2 and therefore,
we find v3 by solving the system
Based on the rref of
, state how many linearly independent eigenvectors can
be generated by
Compute the rref of
To find the eigenvector associated with
, we solve the homogeneous system
for v. This can be achieved by computing the reduced
row echelon form of the matrix
Since we only have one eigenvalue we can simplify the
computation by assigning the value of
Compute the eigenvalues of A by solving the characteristic
and assigning the vector result to
Write your observation about relationship between this
char. eqn and the char. eqn of the scalar differential
equation in part A.
Compute the characteristic equation
Define I as the 3 x 3 identity.
a) finding the
Compare and find the relationship between the solution
of the scalar equation and the solution of the system.
Write your findings below:
The solution of the scalar equation and its derivatives
and the solution of the system are displayed below
for your convenience.
If the two side do not match, it means you have made
a mistake somewhere in your calculations.
Compute the RHS and compare it with the LHS. Use the
Compute the LHS below. Use the simplify
Define the components of the derivative of the solution
as a 3 x 1 vector and name it x'(t).
To verify your solution, you need to compare the left-hand
side x'(t) with the right-hand side Ax(t)
c) Verifying your
Define the general solution (using c1, c2 and c3) one
more time and use the symbolic equal to compute the
particular solution. Use the simplify
Define c1, c2 and c3 using subscripts of s (remember
the subscript starts at 1)
Use the solve command to find c1, c2 and c3 and define
the result as s (see step (b3) in part A).
To find a particular solution, we need to find the values
of c1, c2 and c3 by solving the nonhomogeneous system
. This is done by equating corresponding components
of the general solution evaluated at 0 and the components
of the initial vector b. That is
1. Compute the first and second derivative of the
solution and define them as y'(t) and y''(t), respectively:
b) To find a particular solution satisfying the given
initial condition, we perform the following steps:
Note: Here we have a root with multiplicity 3 and we
can use any element of the vector r to write the solution.
3. Use the roots (elements of the vector r) to write
the general solution
2. Use the solve
command to find the roots of the characteristic equation
and assign the result to r.
Use the coefficients of the differential equation to
write the characteristic equation in terms of the variable
a) To find the general solution we do
a) Find the general solution
b) Find a particular solution satisfying the initial
Given the homogeneous scalar differential equation:
This part of the project will be done for you for comparison
with the results to be obtained by you in part B.
Carefully fill out the personal information at the top
of this worksheet.
Before you start working on the worksheet, use the "save
as" command to save the
worksheet as follows:
xx is your section number
yy is your list number
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You should submit your worksheet in two forms:
a) Softcopy through e-mail
e-mail your worksheet named as explained
in part(8) to
Do not send a zipped file.
Hardcopy (Print out) to be submitted on or before Wednesday
June 2, 2004.
Collaboration is permitted but
copying in punishable by an F grade.
Solve a homogeneous 3rd order scalar differential equation
(the characteristic equation is designed to have a
root with multiplicity 3).
Find a particular solution satisfying given initial
Convert the scalar equation into a system of 3 first-order
Solve the system using the eigenvalue-eigenvector method.
Find a particular solution satisfying initial conditions
that are equivalent to the conditions of the scalar
Verify the solution.
Compare and find the relationship between the components
particular solution of the system and the particular
solution of the scalar equation and its first and second
An individual problem will be generated for you using
your section number, your list number and your student
ID number. Therefore, Section_No, List_No, and Student_ID
are defined above as global variables using the global
". Use the placeholder to enter the numbers.
If you have any question or need help, e-mail me at
or call my mobile 050-384-7098. Good luck.
a) Find the general
Find a particular solution satisfying the initial conditions
where A is (Define your coefficient matrix here) ==>
So, given the system of differential equations
On a piece of paper, convert the differential equation
in part A to a first-order system of the form
and obtain the coefficient matrix A.
A good source to use for help can be found on the WebCT
(this hyperlink will take you to the logon page of
WebCT). Look for the file
"7.5 P25-m3d2.htm" under "Lecture Notes"
on the webCT page of the course. or my
(this will take you directly to the file).
In this part, as stated in the objectives, you will
be asked to convert the scalar differential equation,
solved in part A, to a system of first order differential
equations. Then, solve the system using the eigenvalue-eigenvector
method and compare the two solutions. You will also
be asked to find the relationship between components
of the system solution and the solution of the scalar
equation and its derivatives.
For a reason to be found later, we also compute the
derivatives of the particular solution
4. Substitute the computed values of c1, c2, and c3
into the solution to find a particular solution
Note: The solution is stored in a row vector named s.
Therefore, we can assign the values to c1, c2 and c3
3. Use the solve command to find c1, c2 and c3
2. Evaluate the solution y(t) and its derivatives y'(t)
and y''(t) at t = 0 and set them equal to the initial
conditions. This will give three equations
in the three unknowns c1, c2 and c3 as computed below: