23.3.515 problem 521

Internal problem ID [6229]
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 : 521
Date solved : Friday, October 03, 2025 at 01:57:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.069 (sec). Leaf size: 83

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

\[ y = \frac {\left (i x +1\right )^{\frac {3}{4}+\frac {i}{4}} {\mathrm e}^{-\frac {\arctan \left (x \right )}{2}} \left (\left (x +i\right )^{\frac {1}{2}-\frac {i}{4}} \operatorname {HeunG}\left (2, 1+i, 1, 1, \frac {3}{2}-\frac {i}{2}, 0, -i x +1\right ) c_1 +\left (x +i\right )^{\frac {i}{4}} \operatorname {HeunG}\left (2, \frac {3 i}{2}, \frac {1}{2}+\frac {i}{2}, \frac {1}{2}+\frac {i}{2}, \frac {1}{2}+\frac {i}{2}, 0, -i x +1\right ) c_2 \right )}{\left (x -i\right )^{{1}/{4}}} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

False