# 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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We recall the definition of a root multiplicity. A real or complex number α is called a root of multiplicity k of the
polynomial *p(x)* if there is a polynomial *s(x)* such that \( s(\alpha ) \ne 0 \)
and \( p(x) = \left( x- \alpha \right)^k s(x) . \) If *k=1*, then α is called a simple root.
If \( k \ge 2, \) then α is called a multiple root.

# Reduction of order for constant coefficient equations

We start with constant coefficient linear homogeneous differential equations. Suppose that in a constant coefficient linear differential operator

To find another linearly independent solution to the above homogeneous equation, we use the method of reduction of order, credited to Jacob Bernoulli (1655--1705). Setting

*C*

_{1}and

*C*

_{2}. Finally, substituting for

*v(x)*, we define the general solution

This method can be extended for arbitrary case when a linear constant coefficient differential operator has
a multiple \( \left( \texttt{D} - \gamma \right)^m \) of multiplicity *m*.
Then to such multiple correspond *m* linearly independent solutions:

Theorem: Let \( a_0 , a_1 , \ldots , a_n \) be *n+1* real
(or complex) numbers with \( a_n \ne 0 , \) and *y(x)* be a *n* times
continuously differentiable function on some interval |a,b|. Then *y(x)* is a solution of the *n*-th order
linear differential equation with constant coefficients

We consider a partcular case of second order differential equation---the motion for
undriven damped harmonic oscillator (coefficients μ and ω_{0} > 0 are assumed constants):

**Example:** Let us consider the differential equation

*C*

_{1}and

*C*

_{2}.

soln[x_] = Expand[y[x]/.%[[1]]]

y1[x_] = Coefficient[soln[x],C[1]]

y2[x_] = Coefficient[soln[x], C[2]]

basis = {y1[x], y2[x]}

WronskianDet = Det[{basis, D[basis, x]}]

**Example:** Consider the initial value problem

x'[0] == 5}, x[t], t]

s2[t_] = x[t] /. soln[[1]]

Plot[s2[t], {t, 0.2, 3}]

Out[12]= E^(-3 t) (-1 + 2 t)

Out[13]=

When does the extreme occur?

Where is the extreme?

**Example:** Let us consider the fourth order linear differential operator:

*C*

_{k}, \( k=1,2,3,4 ,\) are arbitrary constants.

# Reduction of order for variable coefficient equations

The reduction of order technique, which applies to arbitrary linear differential equations, allows us to go beyond equations with constant coefficients, provided that we already know one solution. For sake of clarity, we start with a second order linear differential equation with variable coefficients:

*p(x)*and

*q(x)*are some continuous functions on some interval |a,b|. Suppose that we know one its solution \( y = y_1 (x) \ne 0. \) This means that

*v(x)*s determined upon substitution of this form into the given differential equation. First, we use the product rule to obtain

*v(x)*must satisfy the equation

*u(x)*explicitly:

**Example:** Find a second linearly independent solution to the
differential equation

*t*to get:

*u=v'(t)*we have the separable equation

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