> eqn1 := epsilon*diff(y(t), t) - ([y(t)^2 - 1])/2 = 0;
eqn1 := epsilon*diff(y(t), t) + [-y(t)^2/2 + 1/2] = 0
> epsilon := 0.01;
epsilon := 0.01
> f := dsolve({eqn1, y(-1) = 0}, y(t), numeric, output = Array([-1.000, -0.999, -0.995, -0.990, -0.985, 0.980, -0.975, -0.970, -0.965, -0.960]));
Error, (in f) unable to store '[-50.]' when datatype=float[8]

第一个方程里，多写了中括号。你把中括号去掉就能够算了。
eqn1 := epsilon*diff(y(t), t) - (y(t)^2 - 1)/2 = 0;