54.3.246 problem 1262

Internal problem ID [12541]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1262
Date solved : Wednesday, October 01, 2025 at 01:57:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.091 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.186 (sec). Leaf size: 105

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 \exp \left (\int _1^x-\frac {K[1] (K[1]+3)+3}{2 (K[1]+1)^2}dK[1]-\frac {1}{2} \int _1^x\frac {K[2]^2+K[2]-1}{(K[2]+1)^2}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}-\frac {K[1] (K[1]+3)+3}{2 (K[1]+1)^2}dK[1]\right )dK[3]+c_1\right ) \end{align*}

✗ Sympy

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

False