59.1.788 problem 810

Internal problem ID [9960]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 810
Date solved : Wednesday, March 05, 2025 at 08:01:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 22

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

\[ u = {\mathrm e}^{\frac {x^{2}}{2}} c_{1} +c_{2} {\mathrm e}^{\frac {x \left (x +2\right )}{2}} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 24

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

\[ u(x)\to e^{\frac {x^2}{2}} \left (c_2 e^x+c_1\right ) \]

✗ Sympy

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

False