# Preface

This tutorial was made solely for the purpose of education and it was designed for students taking Applied Math 0340. It is primarily for students who have some experience using *Mathematica*. If you have never used
*Mathematica* before and would like to learn more of the basics for this
computer algebra system, it is strongly recommended looking at the APMA
0330 tutorial. As a friendly reminder, don't forget to clear variables in use and/or the kernel. The *Mathematica* commands in this tutorial are all written in bold black font,
while *Mathematica* output is in normal font.

Finally, you can copy and paste all commands into your *Mathematica* notebook, change the parameters, and run them because the tutorial is under the terms of the GNU General Public License
(GPL). You, as the user, are free to use the scripts for your needs to learn the *Mathematica* program, and have
the right to distribute this tutorial and refer to this tutorial as long as
this tutorial is accredited appropriately. The tutorial accompanies the
textbook *Applied Differential Equations.
The Primary Course* by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043

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Introduction to Linear Algebra with

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## Glossary

# Second order systems of equations

Differential equations of arbitrary order with constant coefficients can be solved in straightforward matter by converting them into system of first order ODEs. However, they also can be solved directly, as we demonstrate shortly.

Consider a second order vector differential equation

**A**is a

*n*×

*n*square matrix,

**d**and

**v**are initial displace ments and velocities, and

**x**(t) is a column-vector of

*n*unknown functions to be determined. Its solution

**d**and

**v**are initial displacements and velocities, that is, \( {\bf x}(0) = {\bf d} \) and \( \dot{\bf x}(0) = {\bf v} .\) These two fundamental matrices are solutions of the same matrix differential equation of the second order \( \ddot{\bf P}(t) + {\bf A}\,{\bf P} = {\bf 0}, \) but distinct initial conditions:

**x**(

*t*). Solving the above linear algebraic equation, we obtain

This problem can also be solved by converting the given second order vector differential equation to the system of first order differential equations

*n*column vector of displacements and velocities, and

**B**is a corresponding \( 2n \times 2n \) matrix:

**I**is the identity

*n*×

*n*matrix. The characteristic polynomial for matrix

**B**becomes

Nonhomogeneous second order systems of differential equation with constant coefficients can be solved in straightforward matter based on the Laplace transform. Indeed, consider the initial value problem:

**x**(

*t*) is expressed explicitly:

**f**. This allows us to determine the solution by apllying the inverse Laplace transform:

(which means that all its eigenvalues are positive)

**Example:**

**Example:**Consider the initial value problem for the second order vector differential equation

Eigensystem[A]

**A**has two positive eigenvalues 4 and 25 (such matrices are called positive). ■

**Example: **
■

**Example: **

## Example 2.2.3:

Consider the second order vector differential equation \( \ddot{\bf x} + {\bf A}\,{\bf x} = {\bf 0} ,\) where

1& 4 & 16 \\ 18& 20 & 4 \\ -12 & -14 & -7 \end{array} \right] . \]

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