Fourier Series Applications In Differential Equations - HP 48gII User Manual

Graphing calculator.

Fourier series applications in differential equations

Suppose we want to use the periodic square wave defined in the previous
example as the excitation of an undamped spring-mass system whose
2
2
homogeneous equation is: d
y/dX
+ 0.25y = 0.
We can generate the excitation force by obtaining an approximation with k
=10 out of the Fourier series by using SW(X) = F(X,10,0.5):
We can use this result as the first input to the function LDEC when used to
2
2
obtain a solution to the system d
y/dX
+ 0.25y = SW(X), where SW(X)
stands for Square Wave function of X. The second input item will be the
characteristic equation corresponding to the homogeneous ODE shown above,
i.e., 'X^2+0.25' .
With these two inputs, function LDEC produces the following result (decimal
format changed to Fix with 3 decimals).
Pressing ˜ allows you to see the entire expression in the Equation writer.
Exploring the equation in the Equation Writer reveals the existence of two
constants of integration, cC0 and cC1. These values would be calculated
using initial conditions. Suppose that we use the values cC0 = 0.5 and cC1
= -0.5, we can replace those values in the solution above by using function
SUBST (see Chapter 5). For this case, use SUBST(ANS(1),cC0=0.5) `,
followed by SUBST(ANS(1),cC1=-0.5) `. Back into normal calculator
display we can use:
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