(Solved): Exercise \( 1.3 \) Consider the ordinary differential equation \[ \begin{array}{l} u^{\prime}(t)=t ...

Exercise \( 1.3 \) Consider the ordinary differential equation \[ \begin{array}{l} u^{\prime}(t)=t u(t)(u(t)-2), \\ u(0)=u_{0} . \end{array} \] (a) Verify that \[ u(t)=\frac{2 u_{0}}{u_{0}+\left(2-u_{0}\right) e^{t^{2}}} \] solves (1.56). (b) Show that if \( 0 \leq u_{0} \leq 2 \), then \( 0 \leq u(t) \leq 2 \) for all \( t \geq 0 \). (c) Show that if \( u_{0}>2 \), then \( u(t) \rightarrow \infty \) as \[ t \rightarrow\left(\ln \left(\frac{u_{0}}{u_{0}-2}\right)\right)^{1 / 2} \] (d) Suppose we are interested in (1.56) for \( u_{0} \) close to 1 , say \( u_{0} \in[0.9,1.1] \). Would you say that the problem (1.56) is stable for such data?