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 solu

Question

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.)

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3 weeks 2022-01-01T18:16:02+00:00 1 Answer 0 views 0

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.