Consider the ODE(ordinary differential equation) $$y'=e^{-y^2}+1, y(0)=0.$$ Which of the following statements are true for given ODE?
$P: $ The ODE has unique solution on $\mathbb R.$
$Q:$ The solution of given ODE is bounded on its maximal interval of existence.
According to my analysis, statement $P$ is incorrect because it is an autonomous ODE with the function $f(y) = e^{-y^2} + 1$, and its partial derivative $f_y$ is continuous on the entire real line $\mathbb{R}$. So solution will be unique but no guarantee on full real line. However, I have doubts about statement $Q$. Please provide your suggestions. Thank you.
Best Answer
Hints
(P) Show that the function $u \mapsto 1 + e^{-u^2}$ is Lipschitz continuous on $\Bbb R$, e.g., $e^{-y^2} - e^{-x^2} \leq 2 (y - x)$. Now apply a standard existence–uniqueness theorem.
(Q) Since $y'(t) \geq 1$ for all $t$, for $t \geq 0$ we have $$\int_0^t y'(\tau) \,d\tau \geq \int_0^t 1 \,dt .$$