How can I prove that$$ \left\{\begin{array}{l} y^{\prime}(t)=1-(t+1) e^{y(t)} \\ y(0)=y_0 \end{array}\right. $$ the solution is defined in $[0,+\infty)?$.
My teacher said first to prove that $e^{y(t)} \ge \frac{1}{t+1} \forall t\in $[0,+$T_{max}$)$.$ But how? and second : I know this is the right approach but for me it is not the most natural . Are there faster appproaches?