y=0 is a trivial solution.

Assume y not = 0 (in fact it is zero everywhere or nowhere).

Now divide,

y'/y=t^2

Thus,

INT y'/y dt = INT t^2 dt

ln |y| = (1/3)t^3+C

y=exp[(1/3)t^3 +C]=e^c * exp[1/3t^3]=C*e^{1/3t^3} for C>0.

Note, y=0 was also a solution. Corresponding for C=0.

Thus,

y=C*e^{1/3t^3} for C>=0.

Is the solution on this non-empty open interval.