55.31.16 problem 164

Internal problem ID [13937]
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 : 164
Date solved : Thursday, October 02, 2025 at 08:12:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

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

\[ y = c_1 \left (a +x \right )^{-b +1}+c_2 \left (a -x \right )^{-b +1} \]

✓ Mathematica. Time used: 17.029 (sec). Leaf size: 127

ode=(x^2-a^2)*D[y[x],{x,2}]+2*b*x*D[y[x],x]+b*(b-1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {(x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {(b-1)^2}} (a+x)^{\frac {1}{2}-\frac {1}{2} \sqrt {(b-1)^2}} \left (x^2-a^2\right )^{-b/2} \left (2 a \sqrt {(b-1)^2} c_1 (x-a)^{\sqrt {(b-1)^2}}-c_2 (a+x)^{\sqrt {(b-1)^2}}\right )}{2 a \sqrt {(b-1)^2}} \end{align*}

✗ Sympy

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

False