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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.)
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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 |
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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.
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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.
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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.
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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.
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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.