Exercise 12, § 1

Functions of a Real Variable, IV, 202

Let u be the integral of the equation $x' = \lambda + \frac{x^2}{1 + t^2}$ equal to 0 for $t = 0$ , and let J be the largest interval with left-hand endpoint 0 on which u is continuous.

a) Show that if $\lambda \leq \frac{1}{4}$ one has $J = [0, +\infty[$ (use exerc. 4 of IV, p. 199).

b) Show that if $\lambda > \frac{1}{4}$ one has $J = [0, a[$ where

\sinh \frac{\pi}{2\sqrt{\lambda}} < a < \sinh \frac{\pi}{\sqrt{4\lambda - 1}}

(put

x = y \sqrt{1 + t^2}

and use exerc. 11).

Solution

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