Internal problem ID [9737]
Internal file name [OUTPUT/8679_Monday_June_06_2022_05_09_51_AM_93862035/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}+\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach <- heuristic approach successful <- hypergeometric successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.36 (sec). Leaf size: 250
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)
\[ y \left (x \right ) = x^{\frac {1}{2}+\frac {q}{2}} \left (c_{1} x^{\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {p +q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q +\sqrt {q^{2}+2 q +4 s +1}-\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [1+\frac {\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )+c_{2} x^{-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {p +q -\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, -\frac {-p -q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [1-\frac {\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.338 (sec). Leaf size: 405
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]
\[ y(x)\to c_1 i^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \operatorname {Hypergeometric2F1}\left (\frac {p+q-\sqrt {p^2-2 p-4 r+1}-\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p+q+\sqrt {p^2-2 p-4 r+1}-\sqrt {q^2+2 q+4 s+1}}{2 b},1-\frac {\sqrt {q^2+2 q+4 s+1}}{b},a x^b\right )+c_2 i^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \operatorname {Hypergeometric2F1}\left (\frac {p+q-\sqrt {p^2-2 p-4 r+1}+\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p+q+\sqrt {p^2-2 p-4 r+1}+\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {b+\sqrt {q^2+2 q+4 s+1}}{b},a x^b\right ) \]