3.10.1.2 Example 2

\[ y^{\prime \prime }+xy^{\prime }+y=0 \]

Here \(p_{2}=1,p_{1}=x,p_{0}=1.\) The condition for exactness is

\begin{align*} p_{2}^{\prime \prime }-p_{1}^{\prime }+p_{0} & =0-1+1\\ & =0 \end{align*}

The ode is already exact. i.e. no integrating factor is needed. The solution becomes

\begin{align*} \left ( p_{2}y^{\prime }+\left ( p_{1}-p_{2}^{\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*}