7. THE PHASE LINE AND CLASSIFICATION OF EQUILIBRIA EXERCISES In Exercises 1-3 yo

Question

7. THE PHASE LINE AND CLASSIFICATION OF EQUILIBRIA EXERCISES In Exercises 1-3 you are given a differential equation y' = f(tp), where f satisfies the hypotheses of the Existence and Uniqueness Theorem. You are also given several solutions of this equation and an initial condition. The questionis: using the (Existence and) Uniqueness Theorem, what can you say about the solution y(t) satisfying the given initial condition? 1. yi(t) =-1 and n(t) = 2, for all t, are solutions, initial condition y(0) = 0. 2. yi (t) = t and M2(t) = t +2e-t, for all t, are solutions, initial condition y(0) = 1.

Explanation / Answer

1. Since for all t, y1(t)=-1 and y2(t)=2 are solutions. So for t = 0, they must satisfy the initial condition y(0) = 0.

But when t=0, y1(0) = -1 and y2(0) = 2, which means y1(t) and y2(t) do not satisfy the initial conditions.

2. Here y1(t) = t and y2(t) = t + 2exp(-t) for all t. Again on substituting t = 0 in the expression of y1 and y2, we get

y1(0) = 0 and y2(0) = 2.

But the given initial condition is y(0) =1. So, in this case again y1 and y2 do not satisfy the initial condition.

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