Internal problem ID [8712]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1132.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre]
Solve \begin {gather*} \boxed {2 y^{\prime \prime } x -\left (2 x -1\right ) y^{\prime }+a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 31
dsolve(2*x*diff(diff(y(x),x),x)-(2*x-1)*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \KummerM \left (-\frac {a}{2}+\frac {1}{2}, \frac {3}{2}, x\right ) \sqrt {x}+c_{2} \KummerU \left (-\frac {a}{2}+\frac {1}{2}, \frac {3}{2}, x\right ) \sqrt {x} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 44
DSolve[a*y[x] - (-1 + 2*x)*y'[x] + 2*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {x} \left (c_1 \text {HypergeometricU}\left (\frac {1-a}{2},\frac {3}{2},x\right )+c_2 L_{\frac {a-1}{2}}^{\frac {1}{2}}(x)\right ) \\ \end{align*}