${\displaystyle \epsilon y''+(1+\epsilon )y'+y=0,\,}$

with ${\displaystyle y(0)=0}$, ${\displaystyle y(1)=1}$, for varying ${\displaystyle \epsilon }$.

Created with the following Sage code:

var('t')
var('eps')
y(t) = (exp(-t) - exp(-t/eps))/(exp(-1) - exp(-1/eps))
yO(t) = exp(1 - t)
combined = plot(y.subs(eps=0.1), t, 0, 1, color='red', legend_label='$\epsilon=0.1$') + plot(y.subs(eps=0.07), t, 0, 1, color='green', legend_label='$\epsilon=0.07$') + plot(y.subs(eps=0.04), t, 0, 1, color='blue', legend_label='$\epsilon=0.04$') + plot(y.subs(eps=0.01), t, 0, 1, color='magenta', legend_label='$\epsilon=0.01$') + plot(yO, t, 0, 1, color='black', legend_label='$y_O$')
combined.set_legend_options(shadow=False)
combined.show(ymax=3, axes_labels=['$t$', ''])