# Preface

This is a
tutorial made solely for the purpose of education and it was designed
for students taking Applied Math 0330. It is primarily for students who
have very little experience or have never used
*Mathematica* before and would like to learn more of the basics for this computer algebra system. 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 normal font. 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 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.

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# Operator Methods

Differential equations play a very significant role both in mathematics and in physics because they describe a very wide spectrum of physical phenomena. Therefore, the construction of solutions of differential equations presents the very significant problem. This web site gives an introduction to inverse differential operators.

It is convenient to introduce the notation \( \texttt{D} = {\text d}/{\text d}x \) for the derivative operator. The main problem with definding its inverse is the kernel of the derivative operator which is not zero and it is spanned on constants. From calculus it is known that

*C*, is only the right inverse of the derivative operator:

*f(0) = 0*. In this case, the inverse operator becomes

**I**is the identinty operator.

Our next operator to consider is

*k*is a real number. We define its inverse by solving the differential equation

*C*is an arbitrary constant. For instance,

*C*, we need to impose an initial condition. For example, if we consider the set of functions that vanish at

*x = 0*, we get the inverse operator:

Now we consider the second order differential operator

*k*is a real number. Finding its inverse required application of the variation of parameter and includes two arbitrary constants:

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