81.7.2 problem 8-2

Internal problem ID [21576]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 8. Riccati Equation. Page 124.
Problem number : 8-2
Date solved : Thursday, October 02, 2025 at 07:50:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

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

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

\[ u = {\mathrm e}^{\frac {x^{2}}{2}} \left (c_2 \,\operatorname {erf}\left (\frac {\sqrt {2}\, \left (x -1\right )}{2}\right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 38

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

\begin{align*} u(x)&\to e^x \left (c_1 \operatorname {HermiteH}\left (-1,\frac {x-1}{\sqrt {2}}\right )+c_2 e^{\frac {1}{2} (x-1)^2}\right ) \end{align*}

✗ Sympy

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

False