54.3.52 problem 1057

Internal problem ID [12347]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1057
Date solved : Wednesday, October 01, 2025 at 01:43:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 50

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

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

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 44

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

\begin{align*} y(x)&\to e^{\frac {x^3}{3}+x} \left (c_2 \int _1^xe^{-\frac {1}{3} K[1] \left (K[1]^2+6\right )}dK[1]+c_1\right ) \end{align*}

✗ Sympy

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

False