Casio Classpad fx-CP400: Solving a System of Linear Differential Equations Using Laplace Transforms
Introduction
The following procedure should work with the Casio Classpad family of calculators (300, 330, fx-CP400, fx-CP500).
An attempt to use the Casio Classpad to solve the system of differential equations:
dxf/dt = A * xf(t) + B * yf(t) + g1(t)
dyf/dt = C * xf(t) + D * yf(t) + g2(t)
Initial conditions: xf(0) = xf0, yf(0) = yf0
The independent variable in all the functions is t. The Casio Classpad's CAS functions laplace and invLaplace are used to accomplish this task.
Syntax
Laplace Transform of f(t):
laplace(f(t), t, s)
Laplace Transform of a Differential Equation:
laplace(diff eq, independent variable, dependent variable, s)
Inverse Laplace Transformation:
invLaplace(ℒ(s), s, t)
t: parameter of the original function f(t)
s: parameter of the transformed function ℒ(s)
The definition of a Laplace Transformation:
ℒ(f(t))(s) = ∫( f(t) * e^(-s*t) dt, t = 0 to ∞)
Now, even though we are working a system of differential equations, the laplace and invLapace commands can only work on function at a time. Any other variable will be treated as a constant, even it was meant to represent another functions. The substitution steps will account for this.
The variables used to respect the Classpad's system variables.
Here are the steps I used to solve:
dxf/dt = A * xf(t) + B * yf(t) + g1(t)
dyf/dt = C * xf(t) + D * yf(t) + g2(t)
Initial conditions: xf(0) = xf0, yf(0) = yf0
Spaces are added to readability.
1. Store dxf/dt in the variable f1. Use xf' to represent dxf/dt.
2. Store dyf/dt in the variable f2. Use yf' to represent dyf/dt.
3. Execute the following and substitutions:
laplace( f1, t, xf, s ) ⇒ g1
g1 | xf(0) = xf0 and yf / s = Mp ⇒ g1
4. Execute the following and substitutions:
laplace( f2, t, yf, s ) ⇒ g2
g2 | yf(0) = yf0 and Lp = Mp ⇒ g2
g2 | xf /s = Lp ⇒ g2
Note Lp = ℒ( xf(t) ) and Mp = ℒ( yf(t) )
5. Now solve the system of transformed equations:
solve( {g1, g2}, {Lp, Mp} ) ⇒ lists
6. Take the inverse laplace transforms:
invLaplace( getRight( lists[1], s, t ) ) ⇒ h1
invLaplace( getRight( lists[2], s, t ) ) ⇒ h2
The solutions are:
xf(t) = h1
yf(t) = h2
The getRight command extracts the right side of an equation.
Examples
Example 1:
xf ' = -0.08 * xf + 0.02 * yf + 6
yf ' = 0.08 * xf - 0.08 * yf
Initial conditions: xf(0) = 0, yf(0) = 150
Example 2:
xf ' = 2 * xf + yf + t^2
yf ' = -2 * xf + 5
Initial conditions: xf(0) = 2, yf(0) = 5
Casio Classpad Program: laplaced
Download here: https://drive.google.com/file/d/1zxTEynCjXmBrPU9pYqmXo3kF_so7fiI-/view?usp=share_link
Code:
' setup
SetRadian
' local variables
Local str1, str2, f1, f2
Local g1, g2, lists, h1, h2
Local Mp, Lp, t, s, xval, yval
' main
ClrText
InputStr str1, "xf'(xf,yf,g1(t))="
InputStr str2, "yf'(xf,yf,g2(t))="
Input xval, "xf(0)? "
Input yval, "yf(0)? "
StrJoin "xf'=", str1, str1
StrJoin "yf'=", str2, str2
strToExp(str1) ⇒ f1
strToExp(str2) ⇒ f2
' transform and solve
laplace(f1, t, xf, s) ⇒ g1
g1 | xf(0) = xval and yf/s = Mp ⇒ g1
laplace(f2, t, yf, s) ⇒ g2
g2 | yf(0) = yval and Lp = Mp ⇒ g2
g2 | xf/s = Lp ⇒ g2
Print "Laplace Transforms:"
PrintNatural g1
Print g1
PrintNatural g2
Print g2
solve({g1, g2},{Lp, Mp})⇒lists
invLaplace(getRight(lists[1]),s,t) ⇒ h1
invLaplace(getRight(lists[2],s,t) ⇒ h2
Print "Solutions: "
Print "xf = "
PrintNatural h1
Print h1
Print "yf = "
PrintNatural h2
Print h2
Note: PrintNatural displays the expression in textbook form in a popup box, but it does not add anything to the text output terminal. This is why I included both PrintNatural and Print commands.
Source
Kreyszig, Erwin. Advanced Engineering Mathematics 8th Edition John Wiley & Sons, Inc: New York, NY. 1999. ISBN 0-471-15496-2
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