Trying to solve exercise about Euler method

Question

Help to solve this exercise with Euler Method, my problem here problem is that the solutions I calculated are not close to the real values and they diverge. Thanks!

$$y'=e^y $$

with conditions $$ 0\leq t\leq0.2, y(0)=1$$

First of all, my solution for the ecd was $$y(t)=ln{\frac{e}{1-et}}$$ then for the Euler Method, I used h=0.02, We have the first iteration

$$f(t_i,y_i)=e^y$$ $$y_{i+1}=y_i+f(t_i,y_i)h$$ $$y(0.02)=1+e^{(1)}(0.02)=1.05436$$

t_i	y_real	y_euler
0	1	1
0.02	0.9456	1.05436
0.04	0.8912	1.11177
0.06	0.8369	1.17257
0.08	0.7825	1.23717
0.1	0.7281	1.30609
0.12	0.6738	1.37993
0.14	0.6194	1.45942
0.16	0.565	1.54549
0.18	0.5107	1.63929
0.2	0.4563	1.74233

Answer 1

Please note your $y_{\rm real}$ is not correct:

t_i	y_real	y_euler
0	1	1
0.02	1.0559	1.05436
0.04	1.11511	1.11177
0.06	1.17805	1.17257
0.08	1.24521	1.23717
0.1	1.31722	1.30609
0.12	1.39481	1.37993
0.14	1.47894	1.45942
0.16	1.5708	1.54549
0.18	1.67195	1.63929
0.2	1.78451	1.74233

Calculated using Mathematica:

Table[Log[E/(1 - E* 0.02*i)], {i, 0, 10}]