1.2.5.2.4 Example 4 \(x^{2}y^{\prime \prime }+\left ( x+1\right ) y^{\prime }+y=5\)

\[ x^{2}y^{\prime \prime }+\left ( x+1\right ) y^{\prime }+y=5 \]

Expansion around \(x=x_{0}=0\). This is regular singular point. Hence Frobenius is needed. Comparing the ode to

\[ y^{\prime \prime }+p\left ( x\right ) y^{\prime }+q\left ( x\right ) y=0 \]

Hence \(p\left ( x\right ) =\frac {x+1}{x^{2}},q\left ( x\right ) =\frac {1}{x^{2}}\). Therefore \(p_{0}=\lim _{x\rightarrow 0}xp\left ( x\right ) =\lim _{x\rightarrow 0}\frac {x+1}{x}\) which is not defined. Hence not possible to solve this using series solution.