# [Maxima] manipulation of symbolic expressions

Javed Alam jalam1001 at gmail.com
Tue Mar 17 18:13:52 CDT 2009

```Yes
Here is the solution

http://www.scribd.com/doc/2256846/Symbolic-Math-Paper

Look at the third example and set ZETA=0

More examples for solving differential equations using maxima are here
Linear Differential
Equations<http://tutorial.math.lamar.edu/AllBrowsers/3401/Linear.asp>
Separable Differential
Equations<http://tutorial.math.lamar.edu/AllBrowsers/3401/Separable.asp>Exact
Differential Equations<http://tutorial.math.lamar.edu/AllBrowsers/3401/Exact.asp>
http://www.eng.ysu.edu/~jalam/engr6924s07/sessions/session19/session19.htm

--------------------------------------------------------------------------------------------------------------------------------
Second Order Differential
Equations<http://tutorial.math.lamar.edu/AllBrowsers/3401/SecondOrder.asp>

In the previous chapter we looked at first order differential equations.  In
this chapter we will move on to second order differential equations.  Just
as we did in the last chapter we will look at some special cases of second
order differential equations that we can solve.  Unlike the previous chapter
however, we are going to have to be even more restrictive as to the kinds of
differential equations that we’ll look at.  This will be required in order
for us to actually be able to solve them.

Here is a list of topics that will be covered in this chapter.

*Basic Concepts<http://tutorial.math.lamar.edu/AllBrowsers/3401/SecondOrderConcepts.asp>
*  Some of the basic concepts and ideas that are involved in solving second
order differential equations.

*Real Roots <http://tutorial.math.lamar.edu/AllBrowsers/3401/RealRoots.asp>*
Solving differential equations whose characteristic equation has real
roots.

*Complex Roots<http://tutorial.math.lamar.edu/AllBrowsers/3401/ComplexRoots.asp>
*  Solving differential equations whose characteristic equation complex real
roots.

*Repeated Roots<http://tutorial.math.lamar.edu/AllBrowsers/3401/RepeatedRoots.asp>
*  Solving differential equations whose characteristic equation has repeated
roots.

*Reduction of Order<http://tutorial.math.lamar.edu/AllBrowsers/3401/ReductionofOrder.asp>
*  A brief look at the topic of reduction of order.  This will be one of the
few times in this chapter that non-constant coefficient differential
equation will be looked at.

*Fundamental Sets of
Solutions<http://tutorial.math.lamar.edu/AllBrowsers/3401/FundamentalSetsofSolutions.asp>
*  A look at some of the theory behind the solution to second order
differential equations, including looks at the Wronskian and fundamental
sets of solutions.

*More on the Wronskian<http://tutorial.math.lamar.edu/AllBrowsers/3401/Wronskian.asp>
*  An application of the Wronskian and an alternative method for finding it.

*Nonhomogeneous Differential
Equations<http://tutorial.math.lamar.edu/AllBrowsers/3401/NonhomogeneousDE.asp>
*  A quick look into how to solve nonhomogeneous differential equations in
general.

*Undetermined Coefficients<http://tutorial.math.lamar.edu/AllBrowsers/3401/UndeterminedCoefficients.asp>
*  The first method for solving nonhomogeneous differential equations that
we’ll be looking at in this section.

*Variation of Parameters<http://tutorial.math.lamar.edu/AllBrowsers/3401/VariationofParameters.asp>
*  Another method for solving nonhomogeneous differential equations.

*Mechanical Vibrations<http://tutorial.math.lamar.edu/AllBrowsers/3401/Vibrations.asp>
*  An application of second order differential equations.  This section
focuses on mechanical vibrations, yet a simple change of notation can move
this into almost any other engineering field.

http://www.eng.ysu.edu/~jalam/engr6924s07/sessions/session20/session20.htm

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------

http://tutorial.math.lamar.edu/AllBrowsers/3401/Laplace.asp
<http://tutorial.math.lamar.edu/AllBrowsers/3401/Laplace.asp>
Introduction
to Laplace Transformation

*The Definition<http://tutorial.math.lamar.edu/AllBrowsers/3401/LaplaceDefinition.asp>
*  The definition of the Laplace transform.  We will also compute a
couple Laplace transforms using the definition.

*Laplace Transforms<http://tutorial.math.lamar.edu/AllBrowsers/3401/LaplaceTransforms.asp>
*  As the previous section will demonstrate, computing Laplace transforms
directly from the definition can be a fairly painful process.  In this
section we introduce the way we usually compute Laplace transforms.

*Inverse Laplace
Transforms<http://tutorial.math.lamar.edu/AllBrowsers/3401/InverseTransforms.asp>
*  In this section we ask the opposite question.  Here’s
a Laplace transform, what function did we originally have?

*Step Function<http://tutorial.math.lamar.edu/AllBrowsers/3401/StepFunctions.asp>
*  This is one of the more important functions in the use
of Laplace transforms.  With the introduction of this function the reason
for doing Laplace transforms starts to become apparent.

*Solving IVP’s with Laplace
Transforms<http://tutorial.math.lamar.edu/AllBrowsers/3401/IVPWithLaplace.asp>
*  Here’s how we used Laplace transforms to solve IVP’s.

*Nonconstant Coefficient
IVP’s<http://tutorial.math.lamar.edu/AllBrowsers/3401/IVPWithNonconstantCoefficient.asp>
*  We will see how Laplace transforms can be used to solve some nonconstant
coefficient IVP’s

*IVP’s with Step
Functions<http://tutorial.math.lamar.edu/AllBrowsers/3401/IVPWithStepFunction.asp>
*  Solving IVP’s that contain step functions.  This is the section where the
reason for using Laplace transforms really becomes apparent.

*Dirac Delta Function<http://tutorial.math.lamar.edu/AllBrowsers/3401/DiracDeltaFunction.asp>
* One last function that often shows up in Laplace transform problems.

*Convolution Integral<http://tutorial.math.lamar.edu/AllBrowsers/3401/ConvolutionIntegrals.asp>
* A brief introduction to the convolution integral and an application
for Laplace transforms.

*Table of Laplace
Transforms<http://tutorial.math.lamar.edu/AllBrowsers/3401/Laplace_Table.asp>
*  This is a small table of Laplace Transforms that we’ll be using here.

http://www.eng.ysu.edu/~jalam/engr6924s07/sessions/session21/session21.htm

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Series Solutions to Differential
Equations<http://tutorial.math.lamar.edu/AllBrowsers/3401/Series.asp>We
will finally be looking at non-constant coefficient differential equations.
While we won’t cover all possibilities but one of the common one is
Legendre’s equation shown below:and some additional examples are here
http://www.eng.ysu.edu/~jalam/engr6924s07/sessions/session22/session22.htm

Javed

----------------------------------------------------------------------------------------------------------------------------------------------------------------

On Tue, Mar 17, 2009 at 4:35 AM, Mahery Raharinjatovo <rmaheryl at gmail.com>wrote:

> Hello
>
> I have a question
> Can maxima solve this equation
> 'diff(x,t,2)+k/m*x=0 ?
>
> I try with ode2
>
> (%i1) ode2('diff(x,t,2)+k/m*x=0, x, t);
> but maxima return  this
> "Is  k*m  positive, negative, or zero?"
>