55.29.9 problem 69

Internal problem ID [13842]
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 : 69
Date solved : Thursday, October 02, 2025 at 08:07:37 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x y^{\prime \prime }+a x y^{\prime }+a y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.101 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to e^{-a x} \left (a c_2 x \operatorname {ExpIntegralEi}(a x)-c_2 e^{a x}+c_1 x\right ) \end{align*}

✗ Sympy

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

False