81.7.1 problem 8-1

Internal problem ID [21575]
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 : 8-1
Date solved : Thursday, October 02, 2025 at 07:50:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x u^{\prime \prime }-\left (x^{2} {\mathrm e}^{x}+1\right ) u^{\prime }-x^{2} {\mathrm e}^{x} u&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 28

ode:=x*diff(diff(u(x),x),x)-(x^2*exp(x)+1)*diff(u(x),x)-x^2*exp(x)*u(x) = 0; 
dsolve(ode,u(x), singsol=all);
 

\[ u = \left (\int x \,{\mathrm e}^{-\left (x -1\right ) {\mathrm e}^{x}}d x c_1 +c_2 \right ) {\mathrm e}^{\left (x -1\right ) {\mathrm e}^{x}} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

False