# 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 regular fonts.

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

# Reduction to a single differential equation

The Gauss elimination method for solving systems of algebraic
equations can be adapted to systems of linear differential equations,
not necessarily in normal form (written as \(
\dot{\bf x}(t) = {\bf P}(t)\, {\bf x}(t) + {\bf f}(t) . \) ) In
many cases, it is possible to eliminate all but one dependent variable
in succession until there remains only a single differential equation
containing only one dependent variable. When this single differential
equation can be solved, other dependent variables can be found in
turn, using the original system of equations. Such a procedure, called
the **method of elimination**, provides an effective tool for
solving some systems of differential equations. The solution obtained
may contain the sufficient number of constants of integration to
identify it as the general solution. However, the eliminating
procedure may not lead to an equivalent single differential equation,
and some solutions could be missing. We illustrate the elimination
method in the following examples.

**Example:**
Suppose that there are two large interconnected tanks feeding each
other; one of them we call tank A and the other one tank B. Suppose
that initially tank A holds 160 liters of a brine solution, and tank B
contains 140 liters of the same solution. Fresh water flows into tank
A at a rate 4 liters per minute, and fluid is drained out of tank B at
the same rate. Also, 2 liters per minute of fluid are pumped from
tank B to tank A, and 4 liters per minute from tank A to tank B. The
liquids inside each tank are kept well stirred so that each mixture
is homogeneous. If initially, the brine solution in tank
contains *x*_{0} kg of salt and that tank B
contains *y*_{0} kg of salt, determine the amount of
salt in each tank at time *t* > 0.

**Example:**
Consider again two tanks A and B with capacities 160 and 140 liters,
respectively. Suppose that the rate of exchange between the two tanks
remains the same, at 4 liters per minute. In this case, based on
input and output rates, we obtain the following system of ordinary
differential equations:

**Example:**
Consider the system of ordinary differential equations

*x*

_{1}:

*x*

_{2}is

*c*

_{1}and

*c*

_{2}are arbitrary constants. Using the second equation of the given system

*x*

_{1}:

**Example:**

**Example:**

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