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Laplace + Differential equation solver package version 1.2.4
to TI-89

This package contains functions for solving single or multiple
differential

equations with constant coefficients. Differential equations can
be of any

order and complexity. The functions have also the ability to find
the

solutions of most integral equations or combinations of
differential and

integral equations (integro-differential equations). Method used:
Laplace-

transformation.

This package also contains functions for Laplace-transformation.
If you

already have Laplace92 it can be replaced by this package.

Keep the functions together in a separate folder with the name
"LAPLACE"

and do not create any variable in it.

Functions:

SolveD solve single differential/integral equations

SimultD solve multiple simultaneous differential/integral
equations

Laplace transforms from time to Laplace domain.

iLaplace transforms from Laplace to time domain.

Before using functions set TI-92 MODE

Complex Format to RECTANGULAR

Angle to RADIAN

Exact/Approx to AUTO

You have to do these settings yourself because; the programs
cannot change

the mode setting on the calculator.

(The above statement is not true, if you run menu() the modes
will be set properly.)

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

Help

This program will give online information about and demonstrate
the use of

functions in this package. When you do not need this program any
longer just

delete it.

Syntax: Help()

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SolveD

Solving single differential/integral equation. The Principe in
this function is,

first it will transform the equation in to the Laplace-domain and
second it

solves the equation as a linear equations, third it transforms
the solution back

to the time-domain (see Laplace/iLaplace for further information
about Laplace-

transformation).

In Principe SolveD can solve differential/integral equations of
any order. The

only limitation is the size of the calculator's memory (if it is
a very complex

solution, it may run out of memory).

Equations/initial conditions may contain constants of any kind,
but the letter

's' may not be used in any connection.

Heaviside/Dirac delta functions can be used in equation (see
Laplace for

further information).

Syntax: SolvD

SolvD(equation,{function ,initial conditions})

equation differential/integral equation

derivative of a function is written: d(f(x),x,n)

where "d()" is the normal differentiation function

on the calculator and 'n' is the order.

Integrals of a function is written: 'int'(f(x),x)

or d(f(x),x,-n). Where 'int'() is the calculators

normal integral-function

function function to solve fore: f(x)

initial conditions f(0),f'(0),f''(0),..

Example 1:

Solving second order differential equation:

Equation: d^2x(t)/dt^2+2*dx(t)/dt+5=sin(2*t) and t>=0

Initial conditions: x'=3 and x=1 at t=0

SolveD(d(x(t),t,2)+2*d(x(t),t)+5=e^(-t),{x(t),1,3})

d() is the normal differentiation function on the calculator.

Result: x(t)= 10 - t*e^(-t) - 9*e^(-t) - 5*t +10

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

Example 2:

Obtain the solution x(t), t>=0, of the differential equation

d^2x(t)/dt^2+5*dx(t)/dt+6=f(t)

where f(t) is the pulse function

| 3 (0<= t <6)

f(t) = |

| 0 (t >= 6)

Initial conditions x(0)=0 and x'(0)=2

First rewriting f(t) to Heaviside functions

f(t) = 3*(u(t) - u(t-6))

Now the equation can be solved with SolvD

SolveD( d(x(t),t,2)+5*d(x(t),t)+6=3*(u(t)-u(t-6)),{x(t),0,2})

Result

x(t)=(13/25-3*t/5-13*e^(-5t)/25)*u(t)+(93/25-3*t/5-3*e^(-5*t+30)/25)*u(t-6)

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

Example 3:

Obtain the solution x(t), t>=0, of the integral equation

'int'(x(t),t)+x=sin(5*t)

Where 'int'() is the calculators normal integral-function

SolveD('int'(x(t),t)+x=sin(5*t),{x(t)})

Result

x(t)=5*cos(5*t)/26+25*sin(5*t)/26-5*e^(-t)/26

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

Example 4:

Obtain the solution x(t), t>=0, of the mixed
differential/integral equation

'int'(x(t),t)+dx/dt=cos(t)

Initial conditions: unknown

SimultD('int'(x(t),t)+d(x(t),t)=cos(t),{x(t)})

Solution

x(t)=t*cos(t)/2+x0*cos(t)+sin(t)/2

Here "x0"= the unknown initial condition

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SimultD

Solving multiple simultaneous differential/integral equations.
The Principe in

this function is, first it will transform the equations in to the
Laplace-

domain and second it solves the equations as a system of linear
equations,

third it transforms the solutions back to the time-domain (see
Laplace/iLaplace

for further information about Laplace-transformation).

There are very few rules to obey when using SimultD. First, there
has to be an

equal number of equations and unknown variables. Second, the
variable has to be

a function of the type f(var).

Equations do not need to be of same order. In Principe SimultD
can solve any

number of simultaneous differential/integral equations of any
order or mixture

of different orders, if there are a sufficient number of
equations. The only

limitation is the size of the calculator's memory (if it is a
very complex

solution, it can run out of memory).

Equations/initial conditions may contain constants of any kind,
but the letter 's'

may not be used in any connection.

Heaviside/Dirac delta functions may be used in equations (see
Laplace for

further information).

Syntax: SimultD

SimultD([equation;equation;...],
[f1(var),f1(0),f1'(0),..;f2(var), f2(0),f2'(0),..;.. ])

equation; equation;.. differential/integral equations separated
by ';'

derivative of a function is written: d(f(x),x,n)

where "d()" is the normal differentiation function on

the calculator and 'n' is the order.

Integrals of a function is written: 'int'(f(x),x) or

d(f(x),x,-n). Where 'int'() is the calculators normal

integral-function

f1(var),f1(0),..; f2(var),.. functions and belonging initial
conditions separated

by ';'

Example 1.

Solve for t>=0 the first-order simultaneous differential
equation

dx/dt+dy/dt+5*x+3*y=e^(-t)

2*dx/dt+dy/dt+x+y=3

initial conditions x=2 and y=1 at t=0

[d(x(t),t)+d(y(t),t)+5*x+3*y=e^(-t); 2*d(x(t),t)+d(y(t),t)+x+y=3]
->matx1

SimultD(matx1, [x(t),2;y(t),1])

result

| x(t)=25*e^(t)/3-11*e^(-2*t)-9/2 |

| |

| y(t)=-25*e^(t)/2+1*e^(-t)/2+11*e^(-2*t)/2+15/2 |

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

ATTENTION when solving equations containing integrals. There are
some

situations where Laplace-transformation gives a wrong answer.

1. When an answer from SolveD/SimultD contains Dirac
Delta-functions, it may

indicate, that something is wrong. Use the function Check to see
if the

solution is correct. If Check return something different from
zero the

solution may be false. In most cases just remove the
Dirac-functions, the

rest of the answer will be the correct solution to equation.

2. When the equation contains constants like this:

'int'(f(t),t)+f(t)+sin(t)=const

Laplace-transformation will give the solution for:

'int'(f(t),t)+f(t)+sin(t)=0

Remember when interpreting the results from Check

'int'('Delta'(t),t) = 'Heaviside'(t) = 1

'int'('Delta'(t),t,2) = t*'Heaviside'(t) = t and t>=0

I do actually not know why Laplace-transformation gives these
false solutions.

If somebody knows the explanation/solution to this, I would like
to know it.

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

Check

Function constructed to check that the results from
SolveD/SimultD are correct.

Check will replace the functions in the equations with the output
from

SolveD/SimultD and return the result.

Check always the results from SolveD/SimultD.

Function: Check

Syntax: Check(equation, result from SolveD/SimultD)

equation has to be exactly the same as past to SimultD/SolveD.

result from SolveD/SimultD the second parameter has to be the
result returned

from SolveD/SimultD.

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

Menu

Creates a Custom Menu with functions included in this package.

After executing the program press '2nd' 'CUSTUM' to
activate/deactivate the menu.

Program: Menu

Syntax: Menu()

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Laplace92 version 2.5.1

This packet contains two functions to work with Laplace
transforms and one

Function to solve convolution integrals (folding integrals). All
of them

are able to work with symbols or numbers.

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

Laplace(f(var), var)

Transforms the expression "f(t)" from time domain to
Laplace domain.

f(var): can be any expression, which have a Laplace transform.

var : is the name of the variable to transform normally 't', but
can

be any name.

const: the expression may contain constants of any kind.

Special transforms:

Unit step function (Heaviside function):

Laplace(u(t - a),t) = e^(-a*s)/s

Dirac delta function:

Laplace('delta'(t - a),t) = e^(-a*s)

You can get 'delta' by pressing 'green diamond' + G + D on TI-92.

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

iLaplace(F(var), var):

Transform the other way around from Laplace domain to time
domain.

F(var): can be any polynomial. It may contain ln() and

atan().

var: is the name of the variable to transform normally 's', but
can

be any name.

const: the expression may contain constants of any kind. The
letter 's'

is reserved for the program.

Special transforms:

iLaplace(e^(-a*s)/s,s) = u(t - a)

iLaplace(e^(-a*s),s) = 'delta'(t - a)

iLaplace may never give an error if used correct. It shall be
able to

transform any real polynomial. If this is not the case please
report it to

me.

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fold(f(var), g(var), var)

Solving convolution integrals f*g.

f(var) and g(var): can be any expressions, which has a Laplace
transform.

var: is the name of the variable to integrate normally 't',

but can be any name.

This function use the fact that, if f(t) and g(t) are of
exponential order,

piecewise-continuous on t>=0 and have Laplace transforms F(s)
and G(s)

respectively, then, for Re(s)>0 f*g=InvL{F(s)G(s)}

Example:

f(t) = t*u(t)

g(t) = sin(2t)*u(t)

To solve f*g write following on the commandline:

fold(t*u(t),sin(2t)*u(t),t)

This gives the result: t/2-sin(2t)/4.

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Programs of same type:

Type Name

inverse Z-transformation inverseZ

Please Lars Frederiksen or Roberto
Peres-Franco.

Author: Lars Frederiksen

E-mail:

Supporter on the TI-89 version: Roberto Peres-Franco

E-mail:

Thanks to Roberto Peres-Franco for his great support on the TI-89
version.

PS. Please do not ask for more programs.