1. Consider the initial value problem $$ \frac{d xHd t}=x(\In (x+t)^{2}, \quad x()=e, \quad t \geq 0 $$ Show that the solution blows up to infinity before it reaches $t=1$. You can do it following these steps: (a) Find the solution to the initial value problem \frac{d y}d t}=y(\In (y))^{2}, \quad y)=e, \quad t \geq 0. $$ (6) Show that $y(t) $ blows up to infinity at $t=1$. (c) Using Taylor formula, show that $x(t)>y(t)$ for small $t>0$. (d) Show that if $x(t)>y(t)$ for some value of $t$, then $x^{\prime}(t)>y^{\prime}(+)$. That implies that $xt $ is bigger than $y(t) $ while it exists. SP.SD 475 $$