23.3.417 problem 422

Internal problem ID [6131]
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 : 422
Date solved : Tuesday, September 30, 2025 at 02:22:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 53

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

\[ y = \left (1+x \right ) \left (c_1 \,{\mathrm e}^{-x} \operatorname {HeunD}\left (4, 4, -8, 12, \frac {x}{2+x}\right )+c_2 \operatorname {HeunD}\left (-4, 4, -8, 12, \frac {x}{2+x}\right ) {\mathrm e}^{\frac {-1+x}{2+2 x}}\right ) \]

✓ Mathematica. Time used: 0.386 (sec). Leaf size: 46

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

\begin{align*} y(x)&\to e^{-x} \left (c_2 \int _1^xe^{\frac {K[1]^2+K[1]-1}{K[1]+1}} (K[1]+1)dK[1]+c_1\right ) \end{align*}

✗ Sympy

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

False