2.2.2.3 Example 3

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

Integrating gives\[ y=\frac {x^{2}}{2}+c \] Applying IC gives\[ 1=c \] Hence solution is\[ y\left ( x\right ) =\frac {x^{2}}{2}+1 \]