I am learning Differential Equations, and came across a specific problem of Dirichlet BVP, which says that:
Given x'' = f(x'), x(0) = 0 = x(1), If f(0) $\neq $ 0 and f has two zeros of opposite sign (say, $r^+$ $\gt$ 0 and $r^−$ $\lt$ 0) then all solutions to Dirichlet BVP have derivatives satisfying
$r^−$ $\lt$ x'(t) $\lt$ $r^+$ , $\forall$t $\epsilon$ [0, 1].
Is this true? How can be this shown?
Also, can someone please tell me how to establish apriori bounds on the derivative of solutions on [0, 1] for the BVP x'' = [(x')$^2$ − 1]$^n$ ?