55.29.21 problem 81

Internal problem ID [13854]
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 : 81
Date solved : Thursday, October 02, 2025 at 08:07:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 69

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

\[ y = \frac {{\mathrm e}^{\frac {b^{2}}{2 a}} \pi c_2 \left (a x +b \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, \left (a x +b \right )}{2 \sqrt {a}}\right )+{\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \sqrt {a}\, \sqrt {\pi }\, \sqrt {2}\, c_2 +c_1 \left (a x +b \right )}{x} \]

✓ Mathematica. Time used: 0.186 (sec). Leaf size: 85

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

\begin{align*} y(x)&\to \frac {(a x+b) \left (-\frac {\sqrt {\frac {\pi }{2}} c_2 \text {erf}\left (\frac {a x+b}{\sqrt {2} \sqrt {a}}\right )}{a^{3/2}}-\frac {c_2 e^{-\frac {(a x+b)^2}{2 a}}}{a (a x+b)}+c_1\right )}{b x} \end{align*}

✗ Sympy

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

False