Example 2

\begin{align*} y^{\prime } & =y-1\\ y\left ( 0\right ) & =1 \end{align*}

Solution exists and unique. Integrating gives

\begin{align*} \int \frac {dy}{y-1} & =\int dx\qquad y-1\neq 0\\ \ln \left ( y-1\right ) & =x+c\\ y-1 & =ce^{x}\\ y & =ce^{x}+1 \end{align*}

Applying IC gives

\begin{align*} 1 & =c+1\\ c & =0 \end{align*}

Hence solution is

\begin{align*} y-1 & =0\\ y & =1 \end{align*}