Question

In each of problems 5 through 8, state where in the t y-plane the hypotheses of theorem 2.4.2 are satisfied.

y′ = ( 1 − t² − y²)¹/²

Answer 1

The hypotheses of theorem 2.4.2 are satisfied in the entire t-y plane where the given equation is defined.

Step-by-step explanation:

The given equation is y′ = ( 1 − t² − y²)¹/². The question asks where in the ty-plane the hypotheses of theorem 2.4.2 are satisfied. The hypotheses of theorem 2.4.2 state that the derivative of y with respect to t exists and is continuous on an interval [a, b] and the equation y′ = f(t, y) is continuous on the rectangular region R defined by a ≤ t ≤ b and c ≤ y ≤ d.

By comparing the given equation y′ = ( 1 − t² − y²)¹/² with y′ = f(t, y), we can see that the hypotheses are satisfied in the entire t-y plane where the given equation is defined