55.29.12 problem 72

Internal problem ID [13845]
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 : 72
Date solved : Thursday, October 02, 2025 at 08:07:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 21

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

\[ y = {\mathrm e}^{-a x} \left (c_2 \,x^{-b +1}+c_1 \right ) \]

✓ Mathematica. Time used: 0.153 (sec). Leaf size: 70

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

\begin{align*} y(x)&\to \frac {e^{-a x} x^{\frac {1}{2} \left (-b-\sqrt {(b-1)^2}+1\right )} \left (c_2 x^{\sqrt {(b-1)^2}}+\sqrt {(b-1)^2} c_1\right )}{\sqrt {(b-1)^2}} \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None