61.30.24 problem 172
Internal
problem
ID
[12672]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-5
Problem
number
:
172
Date
solved
:
Tuesday, January 28, 2025 at 08:10:40 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 230
dsolve(x*(x+a)*diff(y(x),x$2)+(b*x+c)*diff(y(x),x)+d*y(x)=0,y(x), singsol=all)
\[
y = c_{2} \left (\operatorname {csgn}\left (a \right ) a +a +2 x \right )^{-\frac {\operatorname {csgn}\left (a \right ) \left (\left (b -2\right ) \operatorname {csgn}\left (a \right ) a +a b -2 c \right )}{2 a}} \operatorname {hypergeom}\left (\left [\frac {\operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right ) a +\sqrt {b^{2}-2 b -4 d +1}\, \operatorname {csgn}\left (a \right ) a -a b +2 c \right )}{2 a}, -\frac {\operatorname {csgn}\left (a \right ) \left (\sqrt {b^{2}-2 b -4 d +1}\, \operatorname {csgn}\left (a \right ) a -\operatorname {csgn}\left (a \right ) a +a b -2 c \right )}{2 a}\right ], \left [-\frac {\operatorname {csgn}\left (a \right ) \left (\left (b -4\right ) \operatorname {csgn}\left (a \right ) a +a b -2 c \right )}{2 a}\right ], \frac {\operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right ) a +a +2 x \right )}{2 a}\right )+c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, -\frac {1}{2}+\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}\right ], \left [\frac {\left (b \,\operatorname {csgn}\left (a \right ) a +a b -2 c \right ) \operatorname {csgn}\left (a \right )}{2 a}\right ], \frac {\operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right ) a +a +2 x \right )}{2 a}\right )
\]
✓ Solution by Mathematica
Time used: 0.249 (sec). Leaf size: 165
DSolve[x*(x+a)*D[y[x],{x,2}]+(b*x+c)*D[y[x],x]+d*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to c_2 a^{\frac {c}{a}-1} x^{1-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\frac {2 c}{a}+\sqrt {b^2-2 b-4 d+1}+1\right ),\frac {b a-\sqrt {b^2-2 b-4 d+1} a+a-2 c}{2 a},2-\frac {c}{a},-\frac {x}{a}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\sqrt {b^2-2 b-4 d+1}-1\right ),\frac {1}{2} \left (b+\sqrt {b^2-2 b-4 d+1}-1\right ),\frac {c}{a},-\frac {x}{a}\right )
\]