# 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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# Reduction Higher Order ODEs

Sometimes higher order differential equations can be reduced to lower and first order equations. We consider two classes of equations when this is possible.

# Dependent Variable Missing

For a differential equation of the form \( y^{(n)} = f(x, y^{(n-1)} ) , \)
the substitution \( p = y^{(n-1)} , \ p' = y^{(n)} \) leads to a first order
equation of the form \( p' = f(x,p) . \) If this equation can be solved for
*p*, then *y* can be obtained by integrating
\( {\text d}^{n-1} y/{\text d}x^{n-1} = p . \) Note that one arbitrary constant
is obtained in solving the first order equation for *p*, and *n-1* constants are introduced
in the integration for *y*.

**Example: ** Consider the differential equation with dependent variable missing:

*p=y'*, we reduce the given differential equation to a linear differential equation of first order

*p = uv*, where

*u*is a solution of the homogeneous equation

*v*we get

*C*

_{1}is a constant of integration. Then we solve for

*p*and obtain

# Independent Variable Missing

Consider second order differential equation of the form \( y'' = f(y,y') , \)
in which the independent variable *t* does not appear explicitly. If we let
\( v = y' = \dot{y} , \) then we obtain
\( {\text d} v/{\text d}t = f(y,v) . \) Since the right side of this equation
depends on *y* and *v*, rather than on *t* and *v*, this equation contains too many
variables. However, if we think of *y* as the independent variable, then by chain rule,
\( {\text d} v/{\text d}t = \left( {\text d} v/{\text d}y \right)
\left( {\text d} y/{\text d}t \right) = v \left( {\text d} v/{\text d}y \right) . \)
Hence the original differential equation can be written as
\( v \left( {\text d} v/{\text d}y \right) = f(y,v). \) Provided
that this first order equation can be solved, we obtain *v* as a function of *y*. A relation
between *y* and *t* results from solving \( {\text d} y/{\text d}t = v(y) , \)
which is a separable equation. Again, there are two arbitrary constants in the final answer.

**Example: ** Consider the differential equation
\( y'' + y \left( y' \right)^3 =0 . \) Upon setting \( y' =v , \)
we reduce the given differential equation of the second order to another one:
\( v \, \frac{{\text d}v}{{\text d}y} + y \left( v \right)^3 =0 . \)
Assuming that \( v \ne 0 , \) we reduce it to the first order equation

**Example: ** Dr. D.H. Parsons derived in 1963

- D.H. Parsons, Exact Integration of the Space-Charge Equation for a Plane Diode: A Simplified Theory, Nature, Volume 200, October, 126--127, 1963.

*V*in a planar vacuum diod

*m*is the mass and

*e*is the magnitude of an electron;

*I*is the current per unit plate area.

The above second order differential equation admits two integrating factors, one less obvious than the other:

*A*and

*B*are arbitrary constants.

In many practical problems, a second order differential equation can be reduced to an exact equation. For example, consider the equation of motion for the damped harmonic oscillator:

_{0}are positive constants. Let

*A*is a constant of motion (which is also called the first integral of the equation). Note that

*A*is similar to the energy integral for a conservatve system. The connection is easily seen if μ = 0, corresponding to the undamped oscilaltor. Following Bohlin

- K. Bohlin, ntegrationsmethode f ̈ur lineare Differential-gleichungen, Astron. Iakttag. Undersokn. Stockholms Observ., Volume 9, 3--6, 1908.

The equation of motion for the damped harmonic oscillator may be derived from the Lagrangian

*q*is

Let us simplify the Hamiltonian with the transformation

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