Using Taylor\'s Series method taking algorithm of order 3, solve the initial value problem y\' = 1 + y; y(0) = 0 with h = 0.5 At x = 1 Solution %% % taylororder 3 rule %% clear all fork=1:2 h=0.5/k ; tmax=1; n=tmax /h; x(1)= 0; y(1)= 0; tout=x(1); yout=y(1).\'; true(1) = 0; ero(1) =0; trout=true(1).\'; erout=ero(1); ero(1) =0; disp(\' x actual Taylor(order3)error \') disp(\'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\') fori=1:(n + 1 ) x(i+1) = x(1)+ i*h; T3 = (1+y(i)+(h/2)*(1+y(i)+x(i) )+h^2*(1+x(i)+y(i)) ); y(i+1) = y(i) + h*T3; % true(i+1) = exp(x(i)) -1; ero(i+1) =abs( y(i) -true(i+1) ); % tout=[tout;x(i)]; yout=[yout,y(i).\']; trout=[trout,true(i).\']; erout=[erout,ero(i).\']; % ifrem(i-1 ,k) ==0% x(i)<= 0.1 %rem(i-1,10) == 0 fprintf(\'%5.2f%12.7f%12.7f%15.7f\ \' ,x(i),true(i+1),y(i), ero(i+1) ); end% if end% i.

1. Using that az + b/cz + d = show that every fractional linear transformation arises as a
combination of horizontal translations z z+b, dilations z az and "reflected inversions" z 1/z.
Conclude that fractional linear transformations preserve angles and the cross-ratio.
Solution
Consider the following operations on z
(z ightarrow cz, z ightarrow z+d, z ightarrow 1/z,z ightarrow (b-ad/c)z,z ightarrow (a/c)+z)
All of these operations are horizontal translations, dilations, or reflected inversions, for some
constants a,b,c and d.
If c is not 0, then apply all the above operations one after another on z:
(z ightarrow cz ightarrow cz+d ightarrow frac{1}{cz+d} ightarrow frac{b-ad/c}{cz+d}
ightarrow frac{a}{c}+frac{b-ad/c}{cz+d}=frac{az+b}{cz+d})
Suppose c is zero.
Then apply the following transforms one after another on z:
(z ightarrow frac{az}{d} ightarrow frac{az}{d}+frac{b}{d}=frac{az+b}{d})
Once again, all of these operations are horizontal translations, dilations, or reflected inversions,
for some constants a,b and d.
Therefore the transform (z ightarrow frac{az+b}{cz+d}) can be obtained as a
combination of horizontal translations, dilations, or reflected inversions.
Since each of horizontal translations, dilations, or reflected inversions preserve angles and cross-
ratios therefore a composition of many such operations also preserve angles and cross-ratios.
Therefore, the fractional linear transform preserves angles and cross-ratios.