Question

MATLAB CODE:

Task 2 8y dt Solve the above ordinary differential equation (ODE) using Euler's method with step sizes of: 2. h 0.75 3. h 0.5 4. h 0.001 a) For each step size, plot the results at each step starting from y(0) 3 to y(3). b) Plot on the same figure as part a) the analytical solution which is given by: 9 24 -8t c) Calculate and print the percentage error between the Euler's method and the analytical result at y(3)

Answer 1

Note: Script is written for h=1. In the same script you can just replace the value of h inorder to get the results. The results have been posted of all h:

clc;clear all;
f=@(t,y) 3*exp(t)-(8*y)/3;
fact=@(t) (9/11)*exp(t)+(24/11)*exp(-8*t/3);
t0=0;
tf=3;
y0=3;
h=1;
t=t0:h:tf;
n=length(t);
t(1)=t0;
y(1)=y0;
yana(1)=y0;
for i=2:n
         y(i)=y(i-1)+h*f(t(i-1),y(i-1));
         yana(i)=fact(t(i));
         err=abs(yana-y);
                    
end
plot(t,y)
hold on
plot(t,yana,'r')
legend('eulers','analytical')
title(['h=',num2str(h)])
err=err(n);
fprintf('For step size of h= %.4f, the error is =%.5f\n',h,err);

1) for h=1

For step size of h= 1.0000, the err r 13 =13.41415 fx >>

Figure 1 File Edit View Insert Tools Desktop Window Help h=1 18 16 14 12 10 eulers analytical 4 -2 2.5 1.5 0.5

2) for h=0.75

Command Window For step size of h= 0.7500, the error is =0.34250 ft >>

Figure 1 File Edit View Insert Tools Desktop Window Help h 0.75 18 16 14 12 10 eulers analytical 0.5 1.5 2.5

3) for h=0.5

Command Window For step size of h= 0.5000, the error is =1.23073 fs >

Figure 1 File Edit View nsert Tools Desktop Window Help h=0.5 18 eulers -analytical 16 14 12 10 2 2.5 1.5 0.5

4) for h=0.001

ommand Window For step size of h- 0.0010, the error is 0.00225

Figure 1 File Edit View Insert Tools Desktop Window Help 稳 h=0.001 18 eulers analytical 16 14 12 10 2.5 1.5 0.5