Let $f \colon \mathbb R \to \mathbb R$ be defined by $ f(y)=\begin{cases} y & \text{ if } y \le 1 \\ 1 & \text{ if } y>1 \end{cases} $
I have to determine explicity the solution $y_a(\cdot)$ of the Cauchy problem $\tag{CP} \begin{cases} y'=f(y) \\ y(0)=a \end{cases} $ where $a>0$.
Well, the function $ y_1(t) = \begin{cases} e^t & \text{ if } t \le 0 \\ t+1 & \text{ if } t > 0 \end{cases} $
works for $a=1$ (it is continuous and differentiable and satisfies (CP) for $a=1$).
Have you got any ideas to treat the case $a\ne 1$? What can we do?