55.28.32 problem 42

Internal problem ID [13815]
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 : 42
Date solved : Thursday, October 02, 2025 at 08:07:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.149 (sec). Leaf size: 70

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

\[ y = {\mathrm e}^{-\frac {x^{2} \left (a \,x^{2}+2 b \right )}{4}} \left (c_1 \operatorname {HeunB}\left (\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right ) x +c_2 \operatorname {HeunB}\left (-\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right )\right ) \]

✓ Mathematica. Time used: 1.291 (sec). Leaf size: 63

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

\begin{align*} y(x)&\to x e^{-\frac {1}{4} x^2 \left (a x^2+2 b\right )} \left (c_2 \int _1^x\frac {e^{\frac {1}{4} \left (a K[1]^4+2 b K[1]^2\right )}}{K[1]^2}dK[1]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq((4*a*x**2 + 2*b)*y(x) + (a*x**3 + b*x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False