Find all explicit solutions y = ϕ (x) of the DE in part (a) defined by this relation. Give the interval I of definition of each explicit solution. (Enter your answers for I using interval notation.)

Answer and Step-by-step explanation:

Firstly, by definition, an interval of definition of any solution is the open interval on which it is defined.

Consider the problem p′=p², p (0) = 1, has solution p (t) = 1 / (1-t).

The definition of this solution is on the interval (-1/2,1/3), making it an interval of definition of solution.

However, it isn't the largest. The largest interval of definition is (-∞,1).

Consider another problem: p (t) = t√ satisfies the ODE p′=1 / (2p).

Meanwhile, when considered as the solution of the ODE above, the domain of square root is [0,∞), and the interval of definition is (0,∞).

But at t=0 the ODE makes no sense, so it would be wrong to assume that y satisfies the ODE in this case. Meanwhile, ODEs are generally considered only on open intervals.