56.2.2 problem 2

Internal problem ID [8806]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 2
Date solved : Sunday, March 30, 2025 at 01:40:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 52

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

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

✓ Mathematica. Time used: 1.281 (sec). Leaf size: 217

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

\[ y(x)\to \frac {1}{2} e^{-\frac {1}{2} (x+2)^2} \left (2 \sqrt {2} e^{\frac {x^2}{2}+x+2} (x+2) \int _1^x\left (\sqrt {2} e^{K[1]} K[1]-e^{-\frac {1}{2} K[1]^2-K[1]-2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {(K[1]+2)^2}}{\sqrt {2}}\right ) K[1] \sqrt {(K[1]+2)^2}\right )dK[1]-\sqrt {2 \pi } \sqrt {(x+2)^2} \left (c_2 e^{\frac {x^2}{2}+x+2}+2 x+2\right ) \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )+2 e^{\frac {x^2}{2}+x+2} \left (2 e^x (x+1)+\sqrt {2} c_1 (x+2)+c_2 e^{\frac {1}{2} (x+2)^2}\right )\right ) \]

✗ Sympy

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

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