Internal problem ID [8865]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1285.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (a -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 52
dsolve(x*(4*x-1)*diff(diff(y(x),x),x)+((4*a+2)*x-a)*diff(y(x),x)+a*(a-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [\frac {a}{2}, \frac {a}{2}-\frac {1}{2}\right ], \relax [a ], 4 x \right )+c_{2} x^{-a +1} \hypergeom \left (\left [1-\frac {a}{2}, -\frac {a}{2}+\frac {1}{2}\right ], \left [2-a \right ], 4 x \right ) \]
✓ Solution by Mathematica
Time used: 0.198 (sec). Leaf size: 144
DSolve[(-1 + a)*a*y[x] + (-a + (2 + 4*a)*x)*y'[x] + x*(-1 + 4*x)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [4]{4 x-1} x^{\frac {1}{2}-\frac {a}{2}} e^{\sqrt {-(a-1)^2} \text {ArcTan}\left (\sqrt {4 x-1}\right )} \left (4 c_1-\frac {c_2 \left (1-i \sqrt {4 x-1}\right )^{-i \sqrt {-(a-1)^2}} \left (1+i \sqrt {4 x-1}\right )^{i \sqrt {-(a-1)^2}}}{\sqrt {-(a-1)^2}}\right )}{2 \sqrt [4]{1-4 x}} \\ \end{align*}