87.14.20 problem 20

Internal problem ID [23539]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 20
Date solved : Thursday, October 02, 2025 at 09:42:48 PM
CAS classification : [_Laguerre]

\begin{align*} x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=1-x \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 24

ode:=x*diff(diff(y(x),x),x)+(1-x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.228 (sec). Leaf size: 27

ode=x*D[y[x],{x,2}]+(1-x)*D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 (x-1) \operatorname {ExpIntegralEi}(x)+c_1 (x-1)-c_2 e^x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (1 - x)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False