Internal problem ID [8808]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1228.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }+a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 59
dsolve((x^2+1)*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (x +\sqrt {x^{2}+1}\right )^{\sqrt {-a +1}}}{\sqrt {x^{2}+1}}+\frac {c_{2} \left (x +\sqrt {x^{2}+1}\right )^{-\sqrt {-a +1}}}{\sqrt {x^{2}+1}} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 66
DSolve[a*y[x] + 3*x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 P_{\sqrt {1-a}-\frac {1}{2}}^{\frac {1}{2}}(i x)+c_2 Q_{\sqrt {1-a}-\frac {1}{2}}^{\frac {1}{2}}(i x)}{\sqrt [4]{x^2+1}} \\ \end{align*}