Internal problem ID [8964]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1385.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Halm]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (a \,x^{2}+a -3\right ) y}{4 \left (x^{2}+1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 61
dsolve(diff(diff(y(x),x),x) = -1/4*(a*x^2+a-3)/(x^2+1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x^{2}+1\right )^{\frac {1}{4}} \left (x +\sqrt {x^{2}+1}\right )^{\frac {\sqrt {-a +1}}{2}}+c_{2} \left (x^{2}+1\right )^{\frac {1}{4}} \left (x +\sqrt {x^{2}+1}\right )^{-\frac {\sqrt {-a +1}}{2}} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 70
DSolve[y''[x] == -1/4*((-3 + a + a*x^2)*y[x])/(1 + x^2)^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {x^2+1} \left (c_1 P_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)+c_2 Q_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)\right ) \\ \end{align*}