MATLAB.4

ASSIGNMENT 4 :

Consider the initial value problem

      z''(x)  + 4*z(x)=0
                   z(0)=1
                   z(0)=-2.

>From Section 5.2 in [B-D] we have a power series representation for z(x)

    z=a0 + a1*x + a2*x^2 + ...

where a0=z(0)=1 and a1=z'(0)=-2.
 
>From Math 262 you can solve the i.v.p. exactly

z=cos(2*x) -sin(2*x)  (for a review of this see Chapter 3 in [B-D]).

First find a2 and a3 by hand (see Section 5.2 in [B-D]). Then make M.files for the three functions

z2(x) = a0 + a1*x + a2*x.^2,    z3(x) = a0 + a1*x + a2*x.^2 +a3*x.^3,

and  z(x) = cos(2*x) -sin(2*x) for x in  [0,pi/2].
 
Plot the three curves on the same figure. Use different line styles
for each curve, and label the figure appropriately.
 
OBSERVE THE APPROXIMATIONS OF Z2(x) AND Z3(x) ON Z(x).