Suppose that x is a solution to the initial value problem x ′ = x 3 − x 1 + t 2 x 2 and x ( 0 ) = 1 2 Show that 0 &lt; x ( t ) &lt; 1 for all t for which x is defined.For this question, I am told to use proof by contradiction. First find constant solutions, and then use intermediate value theorem to argue that since constant solution and initial value solution don't cross, by uniqueness theorem their existence contradict each other, hence 0 &lt; x ( t ) &lt; 1. Can anyone explain how intermediate value theorem plays an effect in here?

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Suppose that   x is a solution to the initial value problem       x    ′    =                    x        3            −      x              1      +              t        2                    x        2             and   x  (  0  )  =      1    2   Show that   0  &amp;lt;  x  (  t  )  &amp;lt;  1 for all   t for which   x is defined.For this question, I am told to use proof by contradiction. First find constant solutions, and then use intermediate value theorem to argue that since constant solution and initial value solution don&#039;t cross, by uniqueness theorem their existence contradict each other, hence   0  &amp;lt;  x  (  t  )  &amp;lt;  1. Can anyone explain how intermediate value theorem plays an effect in here?

Kyan Hamilton · Accepted Answer

Take for example that you somehow found that   x  (  10  )  =  2. As solutions are smooth (at least as smooth as the right side of the ODE), you get by the intermediate value theorem the existence of some point       t    1    ∈  (  0  ,  10  ) with   x  (      t    1    )  =  1. Now assume that       t    1   is minimal. Then in the left neighborhood of       t    1  , the IVP with   x  (      t    1    )  =  1 has two solutions, the non-constant   x with   x  (  0  )  =      1    2   and the constant one. This is a contradiction to the uniqueness claim of the existence-and-uniqueness theorem.

Suppose that x is a solution to the initial value problem x′=(x^3−x)/(1+t^2x^2) and x(0)=1/2 Show that 0<x(t)<1 for all t for which x is defined.

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