61.30.24 problem 172
Internal
problem
ID
[12593]
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
:
Monday, March 31, 2025 at 05:52:40 AM
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*}
✓ Maple. Time used: 0.038 (sec). Leaf size: 230
ode:=x*(x+a)*diff(diff(y(x),x),x)+(b*x+c)*diff(y(x),x)+d*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_2 \left (\operatorname {csgn}\left (a \right ) a +a +2 x \right )^{-\frac {\left (\left (b -2\right ) \operatorname {csgn}\left (a \right ) a +b a -2 c \right ) \operatorname {csgn}\left (a \right )}{2 a}} \operatorname {hypergeom}\left (\left [\frac {\operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right ) a +\sqrt {b^{2}-2 b -4 d +1}\, \operatorname {csgn}\left (a \right ) a -b a +2 c \right )}{2 a}, -\frac {\operatorname {csgn}\left (a \right ) \left (\sqrt {b^{2}-2 b -4 d +1}\, \operatorname {csgn}\left (a \right ) a -\operatorname {csgn}\left (a \right ) a +b a -2 c \right )}{2 a}\right ], \left [-\frac {\operatorname {csgn}\left (a \right ) \left (\left (b -4\right ) \operatorname {csgn}\left (a \right ) a +b a -2 c \right )}{2 a}\right ], \frac {\operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right ) a +a +2 x \right )}{2 a}\right )+c_1 \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, -\frac {1}{2}+\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}\right ], \left [\frac {\left (b \,\operatorname {csgn}\left (a \right ) a +b a -2 c \right ) \operatorname {csgn}\left (a \right )}{2 a}\right ], \frac {\operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right ) a +a +2 x \right )}{2 a}\right )
\]
✓ Mathematica. Time used: 0.249 (sec). Leaf size: 165
ode=x*(x+a)*D[y[x],{x,2}]+(b*x+c)*D[y[x],x]+d*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_2 a^{\frac {c}{a}-1} x^{1-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\frac {2 c}{a}+\sqrt {b^2-2 b-4 d+1}+1\right ),\frac {b a-\sqrt {b^2-2 b-4 d+1} a+a-2 c}{2 a},2-\frac {c}{a},-\frac {x}{a}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\sqrt {b^2-2 b-4 d+1}-1\right ),\frac {1}{2} \left (b+\sqrt {b^2-2 b-4 d+1}-1\right ),\frac {c}{a},-\frac {x}{a}\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
d = symbols("d")
y = Function("y")
ode = Eq(d*y(x) + x*(a + x)*Derivative(y(x), (x, 2)) + (b*x + c)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None