50.2.6 problem 6

Internal problem ID [10096]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 07:02:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-x y^{\prime }-x y-x^{4}-6&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 64

ode:=diff(diff(y(x),x),x)-x*diff(y(x),x)-x*y(x)-x^4-6 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-2-x} \pi c_1 \left (x +2\right ) \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right )-i {\mathrm e}^{\frac {x \left (x +2\right )}{2}} \sqrt {\pi }\, \sqrt {2}\, c_1 +{\mathrm e}^{-x} \left (x +2\right ) c_2 -x^{3}+3 x^{2}-6 x \]

✓ Mathematica. Time used: 2.25 (sec). Leaf size: 271

ode=D[y[x],{x,2}]-x*D[y[x],x]-x*y[x]-x^4-6==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} e^{-x} \left (2 \sqrt {2} (x+2) \int _1^x\frac {1}{2} e^{-\frac {1}{2} K[1]^2-K[1]-2} \left (K[1]^4+6\right ) \left (\sqrt {2} e^{\frac {1}{2} (K[1]+2)^2}-\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {(K[1]+2)^2}}{\sqrt {2}}\right ) \sqrt {(K[1]+2)^2}\right )dK[1]+\left (2 e^{\frac {1}{2} (x+2)^2}-\sqrt {2 \pi } \sqrt {(x+2)^2} \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )\right ) \int _1^x-e^{-\frac {1}{2} K[2]^2-K[2]-2} \left (K[2]^5+2 K[2]^4+6 K[2]+12\right )dK[2]-\sqrt {2 \pi } c_2 \sqrt {(x+2)^2} \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )+2 \sqrt {2} c_1 x+2 c_2 e^{\frac {1}{2} (x+2)^2}+4 \sqrt {2} c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4 - x*y(x) - x*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 6,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**4 - x*y(x) + Derivative(y(x), (x, 2)) - 6)/x cannot be solved by the factorable group method