Internal problem ID [8970]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1393.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {\left (x^{2} b +c x +d \right ) y}{a \,x^{2} \left (x -1\right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 303
dsolve(diff(diff(y(x),x),x) = -(b*x^2+c*x+d)/a/x^2/(x-1)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{\frac {\sqrt {a}+\sqrt {a -4 d}}{2 \sqrt {a}}} \left (x -1\right )^{\frac {\sqrt {a}-\sqrt {a -4 b -4 c -4 d}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [\frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}+\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}, \frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}+\sqrt {a -4 d}-\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [\frac {\sqrt {a}+\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right )+c_{2} x^{\frac {\sqrt {a}-\sqrt {a -4 d}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [\frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}-\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}, \frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}-\sqrt {a -4 d}-\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [\frac {\sqrt {a}-\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right ) \left (x -1\right )^{-\frac {-\sqrt {a}+\sqrt {a -4 b -4 c -4 d}}{2 \sqrt {a}}} \]
✓ Solution by Mathematica
Time used: 0.683 (sec). Leaf size: 46054
DSolve[y''[x] == -(((d + c*x + b*x^2)*y[x])/(a*(-1 + x)^2*x^2)),y[x],x,IncludeSingularSolutions -> True]
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