56.2.47 problem 46

Internal problem ID [8851]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 46
Date solved : Sunday, March 30, 2025 at 01:42:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }-x^{3} y-x^{4}-x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 74

ode:=diff(diff(y(x),x),x)-x^2*diff(y(x),x)-x^3*y(x)-x^4-x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✗ Mathematica

ode=D[y[x],{x,2}]-x^2*D[y[x],x]-x^3*y[x]-x^4-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**4 - x**3*y(x) - x**2*Derivative(y(x), x) - x**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x**2 + x*y(x) + Derivative(y(x), x) + 1 - Derivative(y(x), (x, 2))/x**2 cannot be solved by the factorable group method