55.33.18 problem 227

Internal problem ID [14001]
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 : Friday, October 03, 2025 at 07:23:23 AM
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*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 558

ode:=a*(x^2-1)^2*diff(diff(y(x),x),x)+b*x*(x^2-1)*diff(y(x),x)+(c*x^2+d*x+e)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x^{2}-1\right )^{-\frac {b}{4 a}} \sqrt {2 x +2}\, \left (\left (\frac {x}{2}+\frac {1}{2}\right )^{-\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}-2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [1-\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {x}{2}+\frac {1}{2}\right ) c_1 +\left (\frac {x}{2}+\frac {1}{2}\right )^{\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}-2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [1+\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {x}{2}+\frac {1}{2}\right ) c_2 \right ) \sqrt {2 x -2}\, \left (\frac {x}{2}-\frac {1}{2}\right )^{\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}}}{4} \]

✓ Mathematica. Time used: 131.018 (sec). Leaf size: 1763961

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
e = symbols("e") 
y = Function("y") 
ode = Eq(a*(x**2 - 1)**2*Derivative(y(x), (x, 2)) + b*x*(x**2 - 1)*Derivative(y(x), x) + (c*x**2 + d*x + e)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False