61.32.9 problem 219
Internal
problem
ID
[12640]
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
:
219
Date
solved
:
Monday, March 31, 2025 at 06:49:00 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y&=0 \end{align*}
✓ Maple. Time used: 0.114 (sec). Leaf size: 288
ode:=x^2*(x^2+a)*diff(diff(y(x),x),x)+(b*x^2+c)*x*diff(y(x),x)+d*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x^{\frac {a -c}{2 a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (x^{-\frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {-3 a -c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{4 a}, \frac {-\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}+\left (-2 b +5\right ) a +c}{4 a}\right ], \left [1-\frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_2 +x^{\frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [\frac {3 a +c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{4 a}, \frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}+\left (-2 b +5\right ) a +c}{4 a}\right ], \left [1+\frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_1 \right )
\]
✓ Mathematica. Time used: 1.41 (sec). Leaf size: 336
ode=x^2*(x^2+a)*D[y[x],{x,2}]+(b*x^2+c)*x*D[y[x],x]+d*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to a^{-\frac {\sqrt {a^2-2 a (c+2 d)+c^2}+a-c}{4 a}} x^{-\frac {\sqrt {a^2-2 a (c+2 d)+c^2}-a+c}{2 a}} \left (c_2 x^{\frac {\sqrt {a^2-2 a (c+2 d)+c^2}}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {-2 b a+a+c-\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},\frac {a-c+\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},\frac {\sqrt {a^2-2 (c+2 d) a+c^2}}{2 a}+1,-\frac {x^2}{a}\right )+c_1 a^{\frac {\sqrt {a^2-2 a (c+2 d)+c^2}}{2 a}} \operatorname {Hypergeometric2F1}\left (-\frac {-a+c+\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},-\frac {-2 b a+a+c+\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},1-\frac {\sqrt {a^2-2 (c+2 d) a+c^2}}{2 a},-\frac {x^2}{a}\right )\right )
\]
✗ 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**2*(a + x**2)*Derivative(y(x), (x, 2)) + x*(b*x**2 + c)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None