# Preface

This is a
tutorial 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.

Finally, the commands in this tutorial are all written in bold black font,
while *Mathematica* output is in regular fonts. This means that you can
copy and paste all commands into *Mathematica*, change the parameters and
run them. You, as the user, are free to use the scripts
to your needs for learning how to use 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

Return to computing page for the second course APMA0340

Return to Mathematica tutorial for the first course APMA0330

Return to Mathematica tutorial for the second course APMA0340

Return to the main page for the first course APMA0330

Return to the main page for the second course APMA0340

Return to Part II of the course APMA0340

Introduction to Linear Algebra

## Glossary

# Variation of Parameters

Suppose that we know a fundamental matrix \( {\bf X} (t) \) of the vector system of homogeneous linear differential equations:

**Example:**

# Variation of Parameters for constant matrices

Consider a non-homogeneous vector differential equstion with constant coefficients:

**A**is a given constant matrix and

**f**

*(t)*is a known vector-function.

**Example:**

1/25 E^(-2 t) (-7 + 4 E^(5 t/2) + 3 E^(5 t)),

2/25 E^(-2 t) (-1 + E^(5 t/2))^2}, {-(6/25)

E^(-2 t) (-1 + E^(5 t/2))^2,

1/25 E^(-2 t) (14 + 2 E^(5 t/2) + 9 E^(5 t)),

2/25 E^(-2 t) (-2 - E^(5 t/2) + 3 E^(5 t))}, {-(6/25)

E^(-2 t) (-2 - E^(5 t/2) + 3 E^(5 t)),

1/25 E^(-2 t) (-28 + E^(5 t/2) + 27 E^(5 t)),

1/25 E^(-2 t) (8 - E^(5 t/2) + 18 E^(5 t))}}

Therefore, the matrix **A** has three distinct real eigenvalues: 3, -2, and 1/2

We seek a particular solution in the form:

yp = K0 + K1*t +K2*t^2 + K3*E^t ,

where vector columns K0, K1, K2, and K4 should satisfy the following (algebraic) equations

K1 = A K0 + {0,10,0}

2*K2 = A K1

0 = A K2 + {1,0,0}

K3 = A K3 + {1,0,1}

Let us denote Kj = {k1j,k2j,k3j}, then

Return to Mathematica page

Return to the main page (APMA0340)

Return to the Part 1 Matrix Algebra

Return to the Part 2 Linear Systems of Ordinary Differential Equations

Return to the Part 3 Non-linear Systems of Ordinary Differential Equations

Return to the Part 4 Numerical Methods

Return to the Part 5 Fourier Series

Return to the Part 6 Partial Differential Equations

Return to the Part 7 Special Functions