Internal problem ID [8914]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1335.
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 (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {\left (a x +b \right ) y}{4 x \left (x -1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 57
dsolve(diff(diff(y(x),x),x) = -1/2/x*(3*x-1)/(x-1)*diff(y(x),x)-1/4*(a*x+b)/x/(x-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \LegendreP \left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right )+c_{2} \LegendreQ \left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right ) \]
✓ Solution by Mathematica
Time used: 0.168 (sec). Leaf size: 432
DSolve[y''[x] == -1/4*((b + a*x)*y[x])/((-1 + x)^2*x) - ((-1 + 3*x)*y'[x])/(2*(-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(x-1)^{\frac {-\sqrt {-4 \sqrt {(4 a-1) (a+b)}-8 a-4 b+1} \left (\sqrt {(4 a-1) (a+b)}-2 a\right )+2 b \left (\sqrt {-4 \sqrt {(4 a-1) (a+b)}-8 a-4 b+1}+2\right )+1}{8 b+2}} \left (c_1 \, _2F_1\left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+1\right ),\frac {-4 \sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1} \left (\sqrt {(4 a-1) (a+b)}-2 a\right )+4 b \left (\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+1\right )-\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+1}{16 b+4};\frac {1}{2};x\right )+i c_2 \sqrt {x} \, _2F_1\left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+3\right ),\frac {-4 \sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1} \left (\sqrt {(4 a-1) (a+b)}-2 a\right )+4 b \left (\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+3\right )-\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+3}{16 b+4};\frac {3}{2};x\right )\right )}{\sqrt {1-x}} \\ \end{align*}