2.4.1.3 Example 3 \(y^{\prime }=x\)

\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

\[ y\left ( x\right ) =\frac {x^{2}}{2}+1 \]