y_0 = 2; x_0 = 1; x_final = 20; N = 100;
[x1,y1] = euler_step(y_0,x_0,x_final,N,'f');
N = 2*N;
[x2,y2] = euler_step(y_0,x_0,x_final,N,'f');
N = 2*N;
[x3,y3] = euler_step(y_0,x_0,x_final,N,'f');
N = 2*N;
[x4,y4] = euler_step(y_0,x_0,x_final,N,'f');
N = 2*N;
[x5,y5] = euler_step(y_0,x_0,x_final,N,'f');
N = 2*N;
[x6,y6] = euler_step(y_0,x_0,x_final,N,'f');

figure(1)
clf 
plot(x1,abs(y1-y2(1:2:length(y2))),'b')
hold on
plot(x2,abs(y2-y3(1:2:length(y3))),'k')
plot(x3,abs(y3-y4(1:2:length(y4))),'c') 
plot(x4,abs(y4-y5(1:2:length(y5))),'g')
plot(x5,abs(y5-y6(1:2:length(y6))),'r')

rat1 = (y1 - y2(1:2:length(y2)))./(y2(1:2:length(y2))-y3(1:4:length(y3)));
rat2 = (y2 - y3(1:2:length(y3)))./(y3(1:2:length(y3))-y4(1:4:length(y4)));
rat3 = (y3 - y4(1:2:length(y4)))./(y4(1:2:length(y4))-y5(1:4:length(y5))); 
rat4 = (y4 - y5(1:2:length(y5)))./(y5(1:2:length(y5))-y6(1:4:length(y6)));

figure(2)
clf
plot(x1,rat1,'b')
hold on
plot(x2,rat2,'k')
plot(x3,rat3,'c')
plot(x4,rat4,'g') 
axis([x_0,x_final,0,8])

Y1 = 2*exp(sin(x1))/exp(sin(1));
Y2 = 2*exp(sin(x2))/exp(sin(1));
Y3 = 2*exp(sin(x3))/exp(sin(1));
Y4 = 2*exp(sin(x4))/exp(sin(1));
Y5 = 2*exp(sin(x5))/exp(sin(1));
Y6 = 2*exp(sin(x6))/exp(sin(1));

Rat1 = (y1 - Y1)./(y2(1:2:length(y2))-Y2(1:2:length(Y2)));
Rat2 = (y2 - Y2)./(y3(1:2:length(y3))-Y3(1:2:length(Y3))); 
Rat3 = (y3 - Y3)./(y4(1:2:length(y4))-Y4(1:2:length(Y4)));
Rat4 = (y4 - Y4)./(y5(1:2:length(y5))-Y5(1:2:length(Y5)));
Rat5 = (y5 - Y5)./(y6(1:2:length(y6))-Y6(1:2:length(Y6))); 

figure(3)
clf
plot(x1,Rat1,'b')
hold on
plot(x2,Rat2,'k')
plot(x3,Rat3,'c')
plot(x4,Rat4,'g') 
plot(x5,Rat5,'r') 
axis([x_0,x_final,0,8])