Internal problem ID [8963]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1384.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 73
dsolve(diff(diff(y(x),x),x) = -1/4*(-x^2*(a^2-1)+2*(a+3)*b*x-b^2)/x^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \WhittakerM \left (\frac {b \left (a +3\right )}{2 \sqrt {a^{2}-1}}, \frac {\sqrt {b^{2}+1}}{2}, \sqrt {a^{2}-1}\, x \right )+c_{2} \WhittakerW \left (\frac {b \left (a +3\right )}{2 \sqrt {a^{2}-1}}, \frac {\sqrt {b^{2}+1}}{2}, \sqrt {a^{2}-1}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 96
DSolve[y''[x] == -1/4*((-b^2 + 2*(3 + a)*b*x - (-1 + a^2)*x^2)*y[x])/x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 M_{\frac {(a+3) b}{2 \sqrt {a^2-1}},\frac {\sqrt {b^2+1}}{2}}\left (\sqrt {a^2-1} x\right )+c_2 W_{\frac {(a+3) b}{2 \sqrt {a^2-1}},\frac {\sqrt {b^2+1}}{2}}\left (\sqrt {a^2-1} x\right ) \\ \end{align*}