Stay with the same differential equation:

y'[t] = f[t_,y_] = - t^2 + y;

y[0] = starter

y [t] = 2 - 2 * [ExponentialE]^t + starter * [ExponentialE]^t + 2t + t^2

Use this formula to come up with the exact initial value (starter) on y[0] that gives the break between the two families

----

Yeah, so how do find the exact initial value that'll give the break between a solution that'll graph upwards, and one that'll graph downwards? I'm pretty sure the answer should be 2, but I'm not sure...