23.3.212 problem 214

Internal problem ID [5926]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 214
Date solved : Tuesday, September 30, 2025 at 02:06:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 16

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

\[ y = \left (c_1 \,x^{2}+c_2 \right ) {\mathrm e}^{x}+x \]

✓ Mathematica. Time used: 0.144 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to \frac {1}{2} c_2 e^x x^2+x+c_1 e^x \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2 + x*y(x) + x*Derivative(y(x), (x, 2