Equate the function and its derivatives to the initial
conditions and place them in a vector named z(x).

Now, we use the initial conditions to form the linear
system in for the unknowns c1, c2 ,c3.

Display the gen. soln

Define the general solution

c) To find a particular solution, we first form the
general solution

The Wronskian is never zero since e
3x is never zero. Thus, the solutions are linearly
independent.

The above results can be simplified using the simplify
command

Compute the Wronskian

Define the Wronskian

b) To show that the three solutions are linearly independent,
we first define the Wronskian of the solutions
then compute it

This way you take care of the definition of y1(x) wither
the root is real or complex. Similar definitions can
be given to the other solutions.

Left-hand side is zero which is equal to the right-hand
side.

We can verify our solution by substituting y(x) and
its derivatives in the diff. eqn

Display the particular soln.

You must redefine the general solution again.

Use the values returned in s to define the constants
c1, c2 and c3.

The particular solution is given by

Use the solve command to find c1, c2 and c3. Name the
solution s.

Evaluate the function and its derivatives ant x = 0.

We can display the roots using the subscripts of r as
follows:

Use Mathcad solve
command to find the roots of the char. poly and name
the solution r. Mathcad will return the roots in a
3 x 1 vector.

Define the characteristic poly.

a)

Force Mathcad to start the subscription from 1 rather
than 0.

The solution of this quiz will illustrate how to use
Mathcad to answer the above questions

Solution

Given the differential equation y'' - 3y'' + 25y' +
29y = 0,

a) find three linearly independent solutions to the
diff. eqn.

b) show that the three solutions are linearly independent.
Justify your answer.

c) find a particular solution satisfying the initial
conditions:

y(0) = 20, y'(0) = -40,
and y''(0) = 110

This worksheet should be used as guide for solving the
computer project

Computer
Project Guide

Aug. 12, 2003

Math 260- 023(A Farhat)

You can make your definitions of the solution more general
by creating the definitions in the following manner

Remark:

Display the solutions

Use the above roots to construct the three solutions:

and

The commands Re and Im are used to refer to the real
and imaginary parts of a complex number. In our example,

Remark:

and