Final answer:
Solutions of the equation t'=t-1-y² show that as t increases, the impact of the initial value y is reduced, with t' approaching t-1 minus the square of y. A higher initial y results in a smaller initial t'. The behavior of t' is dependent on the initial value of y when t=0.
Step-by-step explanation:
To understand how solutions behave as t increases for the equation t'=t-1-y², we should consider the relationship between t' and both t and y. The term -y² indicates that as the value of y increases, the quantity being subtracted from t becomes larger, which decreases the value of t'. Hence, for a fixed value of t, a larger initial y value will result in a smaller t'. Conversely, as t increases, the effect of the -y² term becomes less significant, and t' approaches t-1.
Considering when t=0, if the initial value y is 0, then t' will simply be -1. However, if y is nonzero, t' begins at a value less than -1. As t increases, t' will increase as well, but will always be 1 less than t minus the square of y. The behavior of the solutions is ultimately determined by the specific initial value of y.