Final answer:
This question involves solving an initial value problem with a given differential equation and analyzing how the solution's interval of existence depends on the initial condition. The general approach entails variable separation and integration, but the exact solution and interval depend on the specific initial condition provided, which seems to be unclear due to a typographical error.
Step-by-step explanation:
We are asked to solve the given initial value problem and determine the interval in which the solution exists based on the initial value specified. The differential equation provided is dy/dt = t² / (y(1 + t³)), with the initial value y(0) = t0, though t0 seems to be a typo and should possibly be just a specific constant value.
To solve this initial value problem, we would typically separate variables and integrate. However, since the explicit form of the equation and initial conditions are not fully clear due to typographical errors, we can only provide a general approach.
The solution's interval of existence will depend on the initial value, as certain solutions may encounter vertical asymptotes or points of discontinuity within the domain of t, particularly because of the term y in the denominator.