Example 1

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

Solution exists and unique. Integrating gives

\begin{align*} \int \frac {dy}{y} & =\int dx\qquad y\neq 0\\ \ln y & =x+c\\ y & =ce^{x}\end{align*}

Applying IC gives

\[ 1=c \]

Hence solution is

\[ y=e^{x}\]