60.3.386 problem 1392
Internal
problem
ID
[11396]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1392
Date
solved
:
Tuesday, January 28, 2025 at 06:04:51 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }&=-\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.276 (sec). Leaf size: 562
dsolve(diff(diff(y(x),x),x) = -b*x/(x^2-1)/a*diff(y(x),x)-(c*x^2+d*x+e)/a/(x^2-1)^2*y(x),y(x), singsol=all)
\[
y = \frac {\left (x^{2}-1\right )^{-\frac {b}{4 a}} \sqrt {2+2 x}\, \left (c_{1} \operatorname {hypergeom}\left (\left [\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}-2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [-\frac {-2 a +\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {1}{2}+\frac {x}{2}\right ) \left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}}+c_{2} \operatorname {hypergeom}\left (\left [\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}-2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [\frac {2 a +\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {1}{2}+\frac {x}{2}\right ) \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}}\right ) \sqrt {2 x -2}\, \left (-\frac {1}{2}+\frac {x}{2}\right )^{\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}}}{4}
\]
✓ Solution by Mathematica
Time used: 178.035 (sec). Leaf size: 1763961
DSolve[D[y[x],{x,2}] == -(((e + d*x + c*x^2)*y[x])/(a*(-1 + x^2)^2)) - (b*x*D[y[x],x])/(a*(-1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
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