56.2.5 problem 5

Internal problem ID [8809]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 5
Date solved : Sunday, March 30, 2025 at 01:40:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 61

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

\[ y = -{\mathrm e}^{-2-x} \pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right ) c_1 \left (x +2\right )+i {\mathrm e}^{\frac {x \left (x +2\right )}{2}} \sqrt {\pi }\, \sqrt {2}\, c_1 +{\mathrm e}^{-x} \left (x +2\right ) c_2 -x^{2}+2 x -2 \]

✓ Mathematica. Time used: 3.088 (sec). Leaf size: 91

ode=D[y[x],{x,2}]-x*D[y[x],x]-x*y[x]-x^3+2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} e^{-x} \left (-\sqrt {2 \pi } c_2 \sqrt {(x+2)^2} \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )-2 e^x \left (x^2-2 x+2\right )+2 \sqrt {2} c_1 (x+2)+2 c_2 e^{\frac {1}{2} (x+2)^2}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3 - x*y(x) - x*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**3 - x*y(x) + Derivative(y(x), (x, 2)) + 2)/x cannot be solved by the factorable group method