61.28.18 problem 78

Internal problem ID [12499]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 78
Date solved : Thursday, March 13, 2025 at 11:43:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+b \,x^{3} y&=0 \end{align*}

✓ Maple. Time used: 0.719 (sec). Leaf size: 102

ode:=x*diff(diff(y(x),x),x)-(2*a*x+1)*diff(y(x),x)+b*x^3*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{2} \operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{{1}/{4}}}, i \left (-b \right )^{{1}/{4}} x \right ) {\mathrm e}^{a x +\frac {x^{2} \sqrt {-b}}{2}} \left (c_{1} +c_{2} \left (\int \frac {{\mathrm e}^{-x^{2} \sqrt {-b}}}{\operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{{1}/{4}}}, i \left (-b \right )^{{1}/{4}} x \right )^{2} x^{3}}d x \right )\right ) \]

✗ Mathematica

ode=x*D[y[x],{x,2}]-(2*a*x+1)*D[y[x],x]+b*x^3*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(b*x**3*y(x) + x*Derivative(y(x), (x, 2)) - (2*a*x + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False