We consider the Cauchy problem associated with $F_0$ defined by: $$\forall y \in ]0, +\infty[, \quad F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$$ with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$. We denote $\phi_0$ the solution of this problem on $[0, +\infty[$. (a) Show that there exists $\varepsilon > 0$ such that for all $t \in ]0, \varepsilon]$ we have $y_{\text{init}} < \phi_0(t) < \theta$. (b) By considering the function $z_0(t) = \ln\left(\phi_0(t)/\theta\right)$ find the expression of $\phi_0$. (c) Deduce that $\phi_0$ satisfies $y_{\text{init}} < \phi_0(t) < \theta$ for all $t \in ]0, +\infty[$ and that moreover $\phi_0$ is strictly increasing.
We consider the Cauchy problem associated with $F_0$ defined by:
$$\forall y \in ]0, +\infty[, \quad F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$$
with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$. We denote $\phi_0$ the solution of this problem on $[0, +\infty[$.
(a) Show that there exists $\varepsilon > 0$ such that for all $t \in ]0, \varepsilon]$ we have $y_{\text{init}} < \phi_0(t) < \theta$.
(b) By considering the function $z_0(t) = \ln\left(\phi_0(t)/\theta\right)$ find the expression of $\phi_0$.
(c) Deduce that $\phi_0$ satisfies $y_{\text{init}} < \phi_0(t) < \theta$ for all $t \in ]0, +\infty[$ and that moreover $\phi_0$ is strictly increasing.