55.33.9 problem 219

Internal problem ID [13992]
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 : Friday, October 03, 2025 at 07:23:19 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: 0.934 (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]
 

\begin{align*} 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 ) \end{align*}

✗ 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