Internal problem ID [8989]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1410.
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 (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}+\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 253
dsolve(diff(diff(y(x),x),x) = -(a*p*x^b+q)/x/(a*x^b-1)*diff(y(x),x)-(a*r*x^b+s)/x^2/(a*x^b-1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [\frac {p +q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q +\sqrt {q^{2}+2 q +4 s +1}-\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [\frac {b +\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right ) x^{\frac {q}{2}+\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}+\frac {1}{2}}+c_{2} \hypergeom \left (\left [-\frac {-p -q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q -\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [\frac {b -\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right ) x^{\frac {q}{2}-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}+\frac {1}{2}} \]
✓ Solution by Mathematica
Time used: 0.057 (sec). Leaf size: 366
DSolve[y''[x] == -(((s + a*r*x^b)*y[x])/(x^2*(-1 + a*x^b))) - ((q + a*p*x^b)*y'[x])/(x*(-1 + a*x^b)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {i \pi \left (\sqrt {(q+1)^2+4 s}-q-1\right )}{2 b}} a^{\frac {-\sqrt {(q+1)^2+4 s}+q+1}{2 b}} \left (x^b\right )^{\frac {-\sqrt {(q+1)^2+4 s}+q+1}{2 b}} \left (c_2 e^{\frac {i \pi \sqrt {(q+1)^2+4 s}}{b}} a^{\frac {\sqrt {(q+1)^2+4 s}}{b}} \left (x^b\right )^{\frac {\sqrt {(q+1)^2+4 s}}{b}} \, _2F_1\left (\frac {p+q-\sqrt {(p-1)^2-4 r}+\sqrt {(q+1)^2+4 s}}{2 b},\frac {p+q+\sqrt {(p-1)^2-4 r}+\sqrt {(q+1)^2+4 s}}{2 b};\frac {b+\sqrt {(q+1)^2+4 s}}{b};a x^b\right )+c_1 \, _2F_1\left (\frac {p+q-\sqrt {(p-1)^2-4 r}-\sqrt {(q+1)^2+4 s}}{2 b},\frac {p+q+\sqrt {(p-1)^2-4 r}-\sqrt {(q+1)^2+4 s}}{2 b};\frac {b-\sqrt {(q+1)^2+4 s}}{b};a x^b\right )\right ) \\ \end{align*}