60.3.404 problem 1410
Internal
problem
ID
[11414]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1410
Date
solved
:
Tuesday, January 28, 2025 at 06:05:17 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 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 )} \end{align*}
✓ Solution by Maple
Time used: 0.584 (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)
\[
y = x^{\frac {1}{2}+\frac {q}{2}} \left (c_{1} x^{\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {p +q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q +\sqrt {q^{2}+2 q +4 s +1}-\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [1+\frac {\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )+c_{2} x^{-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {p +q -\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, -\frac {-p -q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [1-\frac {\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.248 (sec). Leaf size: 405
DSolve[D[y[x],{x,2}] == -(((s + a*r*x^b)*y[x])/(x^2*(-1 + a*x^b))) - ((q + a*p*x^b)*D[y[x],x])/(x*(-1 + a*x^b)),y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to c_1 i^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \operatorname {Hypergeometric2F1}\left (\frac {p+q-\sqrt {p^2-2 p-4 r+1}-\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p+q+\sqrt {p^2-2 p-4 r+1}-\sqrt {q^2+2 q+4 s+1}}{2 b},1-\frac {\sqrt {q^2+2 q+4 s+1}}{b},a x^b\right )+c_2 i^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \operatorname {Hypergeometric2F1}\left (\frac {p+q-\sqrt {p^2-2 p-4 r+1}+\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p+q+\sqrt {p^2-2 p-4 r+1}+\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {b+\sqrt {q^2+2 q+4 s+1}}{b},a x^b\right )
\]