55.28.25 problem 35

Internal problem ID [13808]
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 : 35
Date solved : Thursday, October 02, 2025 at 08:07:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 79

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

\[ y = \frac {{\mathrm e}^{-\frac {x \left (a \,x^{2}+6 b \right )}{6}} \left (c_2 \,a^{2} x^{3} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {a \,x^{3}}{3}\right )+\frac {5 \left (a \,x^{3}\right )^{{1}/{3}} c_2 3^{{2}/{3}} a \left (a \,x^{3}+2\right ) {\mathrm e}^{-\frac {a \,x^{3}}{6}}}{9}+c_1 \,x^{2} {\mathrm e}^{\frac {a \,x^{3}}{6}}\right )}{x} \]

✓ Mathematica. Time used: 0.163 (sec). Leaf size: 51

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

\begin{align*} y(x)&\to \frac {1}{9} e^{-b x} \left (9 c_1 x-3^{2/3} c_2 \sqrt [3]{a x^3} \Gamma \left (-\frac {1}{3},\frac {a x^3}{3}\right )\right ) \end{align*}

✗ Sympy

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

False