f5. Write the particular solution after finding the coefficients
Define the coefficient as solved for in the previous step.
Place the equations of the system above in a vector and use the "solve" command to find the coefficients.
Equate similar terms to find the linear system involving the undetermined coefficients. This step has to be done manually.
To make it easy to compare similar terms, use the command "collect" on the variable x. This will rearrange the terms as a polynomial in x
Substitute yp (x) and its derivatives in the differential equation and evaluate symbolically.
f4. Substitute yp and its derivatives in the nonhomogeneous equation, simplify, and equate similar terms to determine the coefficients
Define the second derivative as (use literal subscripts) and evaluate symbolically.
Define the first derivative as y'p (x) (use literal subscripts) and evaluate symbolically to display the computed derivative.
f3. Compute the first and second derivative of yp(x)
Define the particular solution as using capital letter A, B, C.. (as needed) for the undetermined coefficients. Use literal subscripts for
f2. Write the modified form of the particular solution yp(x)
f1. Write the unmodified form of the particular solution (skip)
f) Use following steps to find the general solution of the nonhomogeneous equation using the method of undetermined coefficients:
e) Give the interval over which the two solutions of the homogeneous equation are linearly independent (skip)
Compute the determinant of W(x) to find the Wronskian of the two solutions. Use the "simplify" command to simplify the result.
Define the Wronskian matrix as W(x) and evaluate symbolically to display the results.
Define the two solutions as y1 (x) and y2(x) in terms of the roots r1 and r2 , then use the symbolic evaluation to display the two solutions. Use literal subscripts for y and vector subscripts for r.
d) Show that the two solutions of the homogeneous equation are linearly independent. Justify your answer (skip the justification)
Define the complementary solution as and evaluate symbolically to display the solution with the roots substitutedby Mathcad. Use literal subscripts for and the arbitrary constants.
c) Write the complementary solution
Display the roots. Use vector subscripts for r.
Define r as the auxiliary equation and use the "solve" command to find the roots. This way the results will be assigned to r as a vector and the solutions can then be referred to using vector subscripts of r.
b) Find the roots of the characteristic equation
End of Project
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Substitute in the left-hand side of the differential equation and use the "expand" command. The answer should be equal to the right-hand side of the differential equation.
Define y'(x) and y''(x) as the first and second derivative of y(x), respectively and evaluate symbolically to see the computed derivatives.
Check if you got the right answer as follows:
Display the general solution. This will give the particular solution with the constants substituted in the general solution.
Redefine the general solution in terms of and (for the same reason as above.)
Redefine the complementary solution as you did in part (c). This is required by Mathcad since the earlier definition of was given before the c's were defined.
g7) Substitute the above values in the general solution of the nonhomogeneous equation to obtain the particular solution
Define and based on the solution found in the previous step. Use literal subscripts.
g6) Give the values of the constants c1 and c2 found above
Use the "rref" command.
g5) Compute the reduced row echelon form of the augmented matrix to solve the system
g4) Form the augmented matrix of the system above
g3. Write the linear system obtained in parts (g1) and (g2) as a matrix equation A c = b (skip)
Evaluate the first derivative at the initial point and set it equal to the given initial value, then evaluate symbolically to get the second equation of the linear system.
Define the derivative of y(x) as y'(x) and evaluate symbolically.
g2. Compute the first derivative of the general solution in part f6 and apply the second initial condition
Evaluate y(x) at the initial point and set it equal to the initial value, then evaluate symbolically to get the first equation of the linear system involving the arbitrary constants.
g1. Apply the first initial condition on the general solution in part f6.
g) Use the following steps to find a particular solution of the nonhomogeneous equation satisfying the given initial conditions:
Define the general solution y(x) as the sum of the complementary solution and the particular solution and evaluate symbolically to display the result.
f6. Write the general solution of the nonhomogeneous equation
Redefine the particular solution (you may copy it from step (f2)) and evaluate symbolically to display the result.

Note: This redefinition is required by Mathcad so that it can substitute the values of the coefficients.
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Statement of the problem:

Use the method of undetermined coefficients to solve the given initial value problem.
Each student is assigned a different problem. From the list of problems below, select the problem with a number matching your class list number.
Start of Project
Vector Subscripts

Keystroke: [

To type a vector subscript like
1. Type the vector name (v)
2. Type [ (left bracket). The insertion point extends down for typing subscripts
3. Type the subscript (2).
4. Press the spacebar once to get out of subscript mode.

This will refer to the 2nd element of vector v. (provided that the origin is set to 1, otherwise it will refer to the third element of v).
Literal Subscripts

keystroke: period

To type a literal subscript like
1. Type the first part of the name (y)
2. Type a period. The insertion point extends down half a line.
3. Type the second half of the name, the part you want as a subscript (c)
4. Press the spacebar once to get out of subscript mode.

Be sure not to confuse literal subscript with array subscripts . Although they appear identical, they are quite different. Literal subscripts are viewed as an extension of the variable name, while vector subscripts are used to refer to elements of vectors (see below).
Remarks:

1. This computer project is based problem 1 of major exam 2. The problem was designed and solved (see solution of major exam 2) so that it can be used as a template for this project.

2. Some steps given in the solution of the exam problem are omitted in this project. However, to keep a matching reference to the step, they are marked with the word "skip."

3. Instructions and comments are given for each step of the solution. Some instructions are placed in the middle to tell you that the computational result will be long and will require an entire line to display. Thus, you should start typing below the given instruction rather than beside.

4. Mathcad has two types of subscripts: Literal subscripts and Vector vector subscripts. The difference is explained below. You will be guided as to when to use one or the other.
Instructions

1. Carefully fill out the personal information at the top of this worksheet.

2. Before you start working on the worksheet, use the "save as" command to save the
worksheet with the following name:

SxxLyyCP .mcd
where, xx is your section number
yy is your class list number

3. T o facilitate the grading of the project, be as neat as possible and place your input in the space provided for you.

4. Unless absolutely necessary, do not add or delete lines into the document. Enough room is given for writing your answers.

5. If for some reason the screen display gets missed up, press Ctrl+R to refresh the screen.

6. Frequently save your worksheet to avoid losing your work.

7. If you need help with the project or the course material, e-mail me a short description of the problem and I will try to help as soon as possible.

8. You should submit the project on or before Monday January 10, 2005.

e-mail your worksheet named as explained in part (2) above to:
afarahat@kfupm.edu .sa
Do not send a zipped file.
List Number:
Student ID:
Name
Dec. 25, 2004
Computer Project
Math260-041 (A. Farhat)
a) Write the associated homogeneous equation (skip)
<==
Copy the given initial conditions and paste them here.
satisfying the initial conditions
<==
Copy the differential equation you are supposed to solve from the list of problems given above and paste is here.
1. Execute the following steps to find a solution to the differential equation
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