Example 2 \[ y^{\prime \prime }+xy^{\prime }+y=0 \] Here \(p=1,q=x,r=1.\) Let \(F\left ( x,y,y^{\prime },y^{\prime \prime }\right ) =y^{\prime \prime }+xy^{\prime }+y\). The condition for exactness is\[ p^{\prime \prime }-q^{\prime }+r=0 \] Hence the above becomes\begin {align*} 0-1+1 & =0\\ 0 & =0 \end {align*}

The ode is already exact. i.e. no integrating factor is needed. The solution becomes\begin {align*} \left ( py^{\prime }+\left ( q-p^{\prime }\right ) y\right ) ^{\prime } & =0\\ \left ( y^{\prime }+xy\right ) ^{\prime } & =0 \end {align*}

The first integral is\[ y^{\prime }+xy=c_{1}\] Solving this gives\begin {align*} \frac {d}{dx}\left ( Iy\right ) & =Ic_{1}\\ \frac {d}{dx}\left ( ye^{\int xdx}\right ) & =e^{\int xdx}c_{1}\\ ye^{\int xdx} & =\int e^{\int xdx}c_{1}dx+c_{2}\\ y & =e^{\int -xdx}\left ( \int e^{\int xdx}c_{1}dx\right ) +c_{2}e^{\int -xdx}\\ & =c_{1}e^{\frac {-x^{2}}{2}}\left ( \int e^{\frac {x^{2}}{2}}dx\right ) +c_{2}e^{\frac {-x^{2}}{2}dx}\\ & =e^{\frac {-x^{2}}{2}}\left ( c_{1}\int e^{\frac {x^{2}}{2}}dx+c_{2}\right ) \end {align*}