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(where the prime and double primes superscripted functions refer, respec-
tively, to the first and second derivative of this function).

Solution: Introduce the two-dimensional array z, and define

z(1) = y (4.56a)

z(2) = y′ (4.56b)

The system of first-order equations now reads:

z′(1) = z(2) (4.57a)

z′(2) = (1/a)(sin(t) – bz(2) – cz(1)) (4.57b)

Example 4.11
Using the fourth-order Runge-Kutta iterator, numerically solve the same
problem as in Application 1 following Example 4.9.

Solution: Edit and save the following function M-files:

function zp=zprime(t,z)
a=1; b=3; c=1;
zp(1,1)=z(2,1);
zp(2,1)=(1/a)*(sin(t)-b*z(2,1)-c*z(1,1));

The above file specifies the system of ODE that we are trying to solve.
Next, in another function M-file, we edit and save the fourth-order Runge-
Kutta algorithm, specifically:

function zn=prk4(t,z,dt)
k1=dt*zprime(t,z);
k2=dt*zprime(t+dt/2,z+k1/2);
k3=dt*zprime(t+dt/2,z+k2/2);
k4=dt*zprime(t+dt,z+k3);
zn=z+(k1+2*k2+2*k3+k4)/6;

Finally, edit and execute the following script M-file:

yinit=0;
ypinit=0;
z=[yinit;ypinit];

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