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This difference equation will be supplemented in the ODE numerical
solver routine with the iterative equations for D(k) and D2(k), as given respec-
tively by Eqs. (4.16) and (4.30), and with the initial conditions for the function
and its derivative. The first elements for the y, D, and D2 arrays are given by:

y() 1 = y t( = ) 0

dy
D() 1 =
dt t 0=
D () = ( / 1 b() 1 D() 1 − c() 1 y() 1 + u())
a())(−
21
1
1
Application 1
To illustrate the use of the first-order iterator algorithm in solving a second-
order ordinary differential equation, let us find, over the interval 0 ≤ t ≤ 16π,
the voltage across the capacitance in an RLC circuit, with an ac voltage
source. This reduces to solve the following ODE:

2
dy dy
a + b + cy = sin(ω t) (4.36)
dt 2 dt
where a = LC, b = RC, c = 1. Choose in some normalized units, a = 1, b = 3,
ω = 1, and let y(t = 0) = y′(t = 0) = 0.

Solution: Edit and execute the following script M-file:

tin=0;
tfin=16*pi;
t=linspace(tin,tfin,2000);
a=1;
b=3;
c=1;
w=1;
N=length(t);
y=zeros(1,N);
dt=(tfin-tin)/(N-1);
u=sin(w*t);
y(1)=0;
D(1)=0;
D2(1)=(1/a)*(-b*D(1)-c*y(1)+u(1));

for k=2:N

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