\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\[ y=e^{x}\]