61.32.18 problem 227
Internal
problem
ID
[12728]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-7
Problem
number
:
227
Date
solved
:
Tuesday, January 28, 2025 at 08:23:53 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.455 (sec). Leaf size: 562
dsolve(a*(x^2-1)^2*diff(y(x),x$2)+b*x*(x^2-1)*diff(y(x),x)+(c*x^2+d*x+e)*y(x)=0,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: 177.990 (sec). Leaf size: 1763961
DSolve[a*(x^2-1)^2*D[y[x],{x,2}]+b*x*(x^2-1)*D[y[x],x]+(c*x^2+d*x+e)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
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