55.28.23 problem 33

Internal problem ID [13806]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 33
Date solved : Thursday, October 02, 2025 at 08:07:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.153 (sec). Leaf size: 134

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

\[ y = {\mathrm e}^{-\frac {\left (\operatorname {csgn}\left (a \right )-1\right ) a \left (b^{2}-\frac {x^{2}}{3}\right ) x}{2}} \left (c_1 \operatorname {HeunT}\left (-\frac {a 3^{{2}/{3}} b}{\left (a^{2}\right )^{{1}/{3}}}, -6 \,\operatorname {csgn}\left (a \right ), -\frac {a^{2} b^{2} 3^{{1}/{3}}}{\left (a^{2}\right )^{{2}/{3}}}, \frac {3^{{2}/{3}} \left (a^{2}\right )^{{1}/{6}} x}{3}\right ) {\mathrm e}^{\frac {x \,\operatorname {csgn}\left (a \right ) a \left (3 b^{2}-x^{2}\right )}{3}}+\operatorname {HeunT}\left (-\frac {a 3^{{2}/{3}} b}{\left (a^{2}\right )^{{1}/{3}}}, 6 \,\operatorname {csgn}\left (a \right ), -\frac {a^{2} b^{2} 3^{{1}/{3}}}{\left (a^{2}\right )^{{2}/{3}}}, -\frac {3^{{2}/{3}} \left (a^{2}\right )^{{1}/{6}} x}{3}\right ) c_2 \right ) \]

✓ Mathematica. Time used: 2.018 (sec). Leaf size: 55

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

\begin{align*} y(x)&\to \frac {(b-x) \left (c_2 \int _1^x\frac {e^{a b^2 K[1]-\frac {1}{3} a K[1]^3}}{(b-K[1])^2}dK[1]+c_1\right )}{b} \end{align*}

✗ Sympy

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

False